CALYXES AND COROLLAS. Contents 2. Intrinsic Definitions 3

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1 CALYXS AND COROLLAS. DAN YOUNG, H. DRKSN Abstract. A calyx (multiplicative lattice) is a complete lattice endowed with the structure of a monoid such that multiplication by an element is a left adjoint functor of complete lattices (equivalently, a left adjoint functor which preserves colimits). Calyxes are a generalization of the set of ideals of a ring, which form a complete lattice under intersection and summation; in conjunction with the natural multiplication operation (ideal multiplication), this lattice forms a calyx. One can axiomatize the submodules of a module similarly to arrive at the definition of a corolla (module lattice). We present here a notion of principal element equivalent to Dilworth s original definition of principal on a multiplicative lattice, and show that this is equivalent to weak principal. The new concept of principal element begins with a notion of morphism in the category of corollas over a given calyx. We characterize principal elements in the calyx induced by a Noetherian ring as those ideals in the lattice which are principal ideals (in the ordinary sense) after localizing at any given prime ideal. We include here w Contents 1. Intrinsic Definitions 3 2. Intrinsic Definitions Preliminaries Lattices Calyxes Corollas Universal Constructions in Lat, Cal, and -Cor The Category Lat The Category Cal The Category -Cor The Category -Cor Commutative Calyxes and their Corollas Maximal, Minimal, Irreducible, Co-Irreducible, and Prime lements [-1] The Isomorphism Theorems [0] Duality [1] Principal lements [2] xact Sequences [3] Hom and Tensor [10, 11] Deciduous, Coniferous, and Cupulate Calyxes [4] Modularity Perianths Finitely Generated Perianths, Finitely Generated lements Finite Corollas, Finite lements, and Finitely Presented Corollas [5] Calyx Maps of Finite Type and Finite Presentation 34 1

2 2. DAN YOUNG, H. DRKSN Finite Calyx Maps Compact lements and Cupulate Calyxes Localization [9] Base Change [13] The Chinese Remainder Theorem [14] The Spectrum of a Calyx [16] Completions Graded Calyxes and Corollas [55, 57]!!! Noetherian Calyxes [30]!!! Local Calyxes [17] The Nilradical and the Jacobson Radical [18] Nakayama s Lemma [19] Zerodivisors, the Corolla Quotient [24] Supports and Annihilators [39] Valuation Rings [49] Primary Decomposition Dimension [59] Associated Primes [62] Completion [95] Regular Local Rings [105] Schemes Intrinsic Definitions Some xamples Projective Calical Schemes First Properties of Schemes Smooth Schemes Separated and Proper Morphisms Sheaves of Corollas Divisors Projective Morphisms Differentials Formal Calical Schemes Cohomology of Calical Sheaves References 62 In this section we define multiplicative lattices and module lattices from a categorical point of view, providing the necessary insight in establishing a new definition of morphism. This leads to a concept of perennial morphism between module lattices, which is a notion much closer to an actual map of modules. For instance, take a multiplicative lattice. Principal elements of an -module lattice M are in direct correspondence with perennial maps φ : M, where we view the multiplicative lattice as a module lattice over itself. In this document, multiplicative lattices are referred to as calyxes and module lattices are referred to as corollas. This is partially to emphasize the distinction between morphisms

3 CALYXS AND COROLLAS 3 in the category of multiplicative lattices and morphisms in the category of calyxes. Our terminology is taken from the subject of flower anatomy. TODO finitely generated corollas and finitely generated elements. the notation [a 1,..., a n ]. finite as in (a 1,..., a n ). need definition of quotient notation. the free calyx on n generators, free algebra calyx. etc. we write Ω X for the structure sheaf of X. domain calyx open sets U in X change name to calical schemes change abstract tensor universal property finite module is different than a finite element. section on modularity move annihilators from Zerodivisors, the Corolla Quotient to section on associated primes fix graded corollas. remember finitely generated. check the hom functor fix primary decomposition localization at a principal element notation. note that it is possible to fully recover a spectrum from its caylx. compact elements 1. Intrinsic Definitions Let C be a the category set and the category of sup-lattices -there is a left adjoint functor set to sup-lattices, injective on objects -a set can be recovered from its sup-lattice the category monoid and the category of calyxes -ab = xy : x in a, y in b sumd : ad = c - c b - ab -xy in c for all x in a and y in b iff x in b implies ax = c thus every monoid induces a calyx -the functor monoid to calyx is left adjoint. the category of group actions and the category of corollas the category of rings and the category of perianths the category of modules and the category of sepals 2.1. Preliminaries. 2. Intrinsic Definitions Definition 0. A lattice is a poset for which every two elements x, y have a supremum and an infimum. A bounded lattice is a lattice which has two elements 0, 1 such that 0 x 1 x. A complete lattice is a lattice such that any arbitrary collection of elements has a unique infimum and supremum. Any complete lattice is bounded. Meets and joins in a lattice are automatically associative, commutative, and idempotent. Lemma 1. Suppose a poset P has arbitrary suprema. Then P has arbitrary infima, so that P is a complete lattice. Proof. Take a subset X P and consider the set Y = {y P : y x x X} of lower bounds of X. This set has a supremum p P. Take x X. y x y Y, so p x. Since x was chosen arbitrarily, p Y. But by definition p y y Y, so p is an infimum for X. Lemma 2. Let f : P Q be a morphism of complete lattices which preserves suprema (colimits). Then f has a right adjoint.

4 4. DAN YOUNG, H. DRKSN Proof. Define a function g : Q P where g sup{x P : f(x) y}. If x P, y Q are such that f(x) y, then x sup{x P : f(x) y} = g(y). Conversely, if x g(y), then f(x) f(g(y)) = f(sup{x P : f(x) y}) = sup f({x P : f(x) y}) y Lemma 3. It follows from the proof above that, for an adjoint pair f g of complete lattice morphisms f : P Q and g : Q P, f(x) = lim{y Q : x g(y)} and g(y) = lim{x P : f(x) y}. Lemma 4. Let F : C D be a left adjoint functor with right adjoint G : D C. Let A X,Y : Hom D (F (X), Y ) Hom C (X, G(Y )) be the isomorphisms testifying to the adjoint relationship F G for each X C and Y D. We show that F preserves colimits. Proof. Let Λ be a small category and Φ : Λ C a functor such that lim Φ exists. Let α λ : Φ(λ) lim Φ be the canonical maps. We show that {F (lim Φ), F (α λ )} forms a colimit for F Φ. Take an object P D with morphisms β λ : F Φ(λ) P such that the following diagram commutes for each φ : λ µ in Λ: F Φ(λ) F (φ) F Φ(µ) β λ β µ Since F G the following diagram commutes: P 1 P P Φ(λ) A(β λ ) φ Φ(µ) A(β µ) 1 G(P ) G(P ) G(P ) By the universal property of colimit, there is β : lim Φ G(P ) such that the following diagram commutes: Φ(λ) Φ(µ) lim Φ A(β λ ) φ A(β µ) α µ 1 G(P ) 1 G(P ) G(P ) G(P ) G(P ) Since F G the following diagram commutes: β φ F Φ(λ) F Φ(µ) F (lim Φ) β λ β µ A 1 (β) α µ 1 G(P ) 1P P P P We can apply this argument in reverse to get uniqueness. Analogously, right adjoints preserve limits. Corollary 5. Colimits commute and limits commute.

5 CALYXS AND COROLLAS 5 Proof. Follows since colimits are left adjoint and limits are right adjoint. A proof can be found in A Term of Commutative Algebra By Allen Altman and Steven Kleiman. It occurs as 6.6. Definition 6 (Composition of Adjoints). Let f = (f, f ) : C D be an adjoint pair with f : C D, f : D C. Suppose g = (g : g ) : D is another adjoint pair with g : D and g : D. We can form the composition g f = (g f, f g ) of the two adjoints, which is an adjoint in its own right. We check this: Hom (g f (X), Y ) = Hom D (f (X), g (Y )) = Hom C (X, f (g (Y ))) Moreover the identity functor on a category C composes with any adjoint pair f to make f again. In this way, one can replace the functors in Cat with adjoint pairs to obtain a category. This same process works with the category of posets viewed as a full subcategory of Cat Lattices. To motivate the definition of a calyx we start with the notion of a lattice induced by a monoid. We form the corresponding category of complete lattices with morphisms adjoint pairs of poset functors. From complete lattices we construct calyxes just as rings are constructed from abelian groups. Definition 7. Let M be a (not necessarily commutative) monoid and consider the lattice = (, +, ) of its submonoids, where + is the join and is the meet. The lattice is complete, with M and 0 the bounds. To each monoid M we assign the lattice of its submonoids by M lat. Note that, while the monoid M may not be commutative in general, joins and meets in M lat are still commutative. lat can be made into a functor in a natural way, which is desirable as it vital to the other definitions to come, but this cannot happen on the ordinary category of lattices and lattice morphisms. To make functorial, let us examine the properties of extension and contraction of submonoids under a morphism f : M N of monoids. Write K e (N) for the image (extension) of K (M) and L c (M) for the preimage (contraction) of L (N). Take K, K, {K i } (M) and L, L, {L i } (N). Then K K ec L ce L K ( K K e K e L L L c L c K ) e i = Ke i Lc i = ( L i ( K i) e Ke i Lc i ( L i 0 e = 0 1 c = 1 These properties can be seen to follow from the fact that extension and contraction are a pair of adjoint poset functors, where the lattices in question are viewed as poset categories. We denote by Lat the category of complete (bounded) lattices which has as its morphisms adjoint pairs φ = (φ, φ ), φ φ with φ : M N and φ : N M, written φ : M N. An unfortunate consequence of this is that the left adjoint φ is written left to right. Note the distinction between Lat and the ordinary category of lattices and lattice morphisms. In the second, morphisms are not adjoint pairs but maps f : P Q such that x y f(x) f(y). The composition of an adjoint pair is an adjoint pair and there is the identity adjoint pair on each complete lattice, so Lat indeed forms a category. We write φ for the ) c ) c

6 6. DAN YOUNG, H. DRKSN left adjoint and φ for the right adjoint of a morphism pair φ : M N in Lat, with φ written left to right and φ right to left. xample 8. Let be a complete lattice. The set nd Lat () of complete lattice morphisms forms a monoid under composition. nd Lat () also has a canonically induced partial order where φ ψ when φ (x) ψ (x) x and φ (x) ψ (x) x. Under this order nd Lat () is bounded and complete. To show this it suffices to show that nd Lat () has arbitrary joins, by lemma 1. Let {φ i } be elements of nd Lat (). Define α (x) = (φ i) (x) and α (x) = (φ i) (x). α and α are poset functors ; since colimit is a functor, x y (φ i ) (x) (φ i ) (y) since limit is a functor, x y (φ i ) (x) (φ i ) (y) To show α α we must show (φ i) (x) y x (φ i) (y). Observe that (φ i ) (x) y (φ i ) (x) y i I x (φ i ) (y) i I x (φ i ) (y) A 0 element for F is (φ, φ ) where φ (x) = 0 for each x and φ (x) = 1 for each x. A 1 element for F is (ψ, ψ ) where ψ (x) = x for each x and ψ (x) = x for each x Calyxes. Remark. Notice that a ring A is an abelian group G with a group homomomorphism φ : G nd Ab (G) such that φ(1) = 1 and φ(φ(x)(y))(z) = φ(x)(φ(y)(z)) x, y, z G. We can use this as motivation in constructing the category of calyxes from the category of complete lattices. Definition 9. A calyx is a complete lattice along with a complete lattice morphism φ = (φ, φ ) : nd Lat () such that ψ(φ (a)(b))(c) = ψ(a)(ψ(b)(c)) where ψ = φ or φ (called associativity of φ) and φ(1) = 1. We denote the join in with summation notation and the meet with intersection notation. We denote φ (a)(b) by ab for a, b (called product) and φ (a)(b) by (b : a) (called quotient), to match the notation for product and ideal quotient. It follows from the axioms that φ : nd Lat () is a morphism of monoids (with the product in being (a, b) ab and the product in nd Lat () being composition of adjoint pairs). A calyx is called commutative if φ (a)(b) = φ (b)(a) a, b.

7 CALYXS AND COROLLAS 7 In light of the following motivating example, the theory of calyxes can be viewed as a generalization of ring theory: xample 10. The set of ideals of any commutative ring A forms an example of a complete lattice A cal, whose meet is intersection and whose join is ideal sum. We define the left adjoint φ : (φ, φ ) : A cal nd(a cal ) where φ (a)(b) = ab (the ideal product) and φ (a)(b) = (b : a) (the ideal quotient), and this makes into a calyx. Proof. Take a A cal. For each b, c A cal, b (c : a) ab c, so φ (a) φ (a). The map φ forms the left adjoint of a lattice morphism pair. We check this as follows: if a b and c A cal then ac bc and (c : b) (c : a). Suppose {a i } and b are elements of A cal. Then φ ( ) ( ) a i (b) = a i b = (a i b) = ( lim φ (a i ) ) (b) φ (a i )(b) = ( b : a i ) = (b : a i ) = ( lim φ (a i ) ) (b) So that φ preserves colimits. By lemma 2, φ is indeed left adjoint. Lemma 11. ab = b (c:a) c and (a : b) = cb a c for elements a, b, c of any calyx. Proof. ab x b (x : a). So y = ab y x ab x y = inf{x : ab x} = inf{x : b (x : a)} The other claim follows similarly. Definition 12. We often write µ a (b) for ab in a calyx. We write µ a(b) for (b : a). This is to emphasize that µ a is an element of nd Lat (). Lemma 13. The following familiar properties of ideals in ring theory hold in a general for an arbitrary calyx with left adjoint structure map φ = (φ, φ ) : nd(). Let a, b, c, {a i } be elements of.

8 8. DAN YOUNG, H. DRKSN Reason Property Dual property φ(1) = 1 1a = a (a : 1) = a Associativity of φ (ab)c = a(bc) (a : b) : c) = (a : bc) φ is a functor a b ac bc a b (c : a) (c : b) φ (a) and φ (a) are functors. b c ab ac b c (b : a) (c : a) φ (a) and φ (a) are functors ab a a (a : b) and b 1 Since φ is left adjoint to φ, a φ φ (a) and φ φ (a) a (a : b)b a a (ab : b) φ(a) is an adjoint pair so φ (a) (resp. φ (a)) distributes over coproducts (resp. products). φ is an adjoint pair so φ (resp. φ ) preserves initial objects (resp. terminal objects) Canonical morphism from universal property of product (resp. coproduct) φ is itself left adjoint, and so distributes over colimits. φ is left adjoint and so sends initial objects to initial objects. (aa i) = a ( a ) i (a i : a) = ( a i : a ) a0 = 0 a ( a ) i aa i ( a ) i a = (a ia) (a : 1) = a 0a = 0 (a : 0) = 1 The following identities follow from the properties listed above: (a + b)(a b) ab a + b (ab : a b) (a : a + b) = (a : b) ab a b a + b = 1 a b = ab a b (b : a) = 1 a + c = 1, a + b = 1 a + (b c) = b + c (a b : b) = (a : b) a + b = 1, a + c = 1 (ab) + c = 1 (a : a i) ( a : a ) i ( a : a ) i = (a : a i) The list of properties tabulated above determines the definition of a calyx, but it is clearly not minimal. To satisfy those who would like a minimal set of axioms, we show that this is equivalent to a multiplicative lattice. Lemma 14. Let be a complete lattice on which there is a binary operation called product, written (a, b) ab, such that the following hold for each a, {a i } : (1) Product forms an abelian monoid. (2) a a i = aa i (3) a0 = 0 We call a multiplicative lattice. Obviously each calyx is a multiplicative lattice. We show that each multiplicative lattice is a calyx.

9 CALYXS AND COROLLAS 9 Proof. For each a define µ(a) : to be the adjoint pair whose left adjoint µ(a) is multiplication by a. Indeed, multiplication by a is left adjoint by the adjoint functor theorem for posets since it distributes over sums and preserves the initial object. This defines a set map µ : nd Lat () which again distributes over the initial object and sums. Thus µ is itself a left adjoint. Since multiplication is associative, the left adjoint of the maps below match, so that they are equal: µ(µ(a) (b))(c) = µ(a)(µ(b)(c)) a, b, c Also µ(a) (b) = µ(b) (a), so that µ(a) (b) = µ(b) (a) for each a, b. calyx. Thus is a xample 15. Let = Z cal, the calyx over the integers. Then + and distribute over each other, so that is a distributive lattice. This is not true in general. The best one obtains in this direction in general is the modular law, which holds for any calyx induced by a ring: If b a or c a then a (b + c) = a b + a c Definition 16. A morphism of calyxes and F is a set map φ : F which is both a morphism of complete lattices on the underlying complete lattice structure and a morphism of monoids on the underlying monoid structure. We denote the category of calyxes and calyx morphisms by Cal. Lemma 17. There is a functor Cal Mon. For a calyx define a monoid M with the same underlying set as, where multiplication µ : M M M in M is defined by (a, b) ab. Indeed, M has 1 as an identity element and is associative by requirement on. There is also a forgetful functor Cal Lat which forgets all but the lattice structure. Definition 18. We call a calyx representable if = A cal for some ring A. Not every calyx is representable. For example, take a semiring S over the group D 8 of symmetries of a square. The lattice structure of D 8 is nonmodular, and a semiring over D 8 exists whose two sided ideal structure is nonmodular. One can construct a calyx from S in the same way as with rings, forming in particular a lattice with non-modular ideal structure. Wheras every calyx arising from a ring has modular ideal structure. One might ask if the representable calyxes are exactly the modular ones. Lemma 19. The following familiar properties of extension and contraction of ideals hold in general for an arbitrary calyx morphism f : F. Let a, a, a, {a i } be elements of the calyx and let b, b, b, {b j } j J be elements of the calyx F.

10 10. DAN YOUNG, H. DRKSN Reason Property Dual Property f is a monoid morphism and f (1) = 1 so sends 1 to 1 f is a monoid morphism and f (aa ) = f (a)(a ) so distributes over product. Lemma 20 f (a)f (a ) f (aa ) f (b)f (b ) f (bb ) Lemma 20 f ((a : a )) (f (a) : f (a )) f ((b : b )) (f (b) : f (b )) f (resp. f ) is left adjoint f ( a ) i = f (a i ) f ( b ) i = f (b i ) (resp. right adjoint) and so distributes over colimits (resp limits) f (resp. f ) is left adjoint f (0) = 0 f (1) = 1 (resp. right adjoint) and so preserves initial objects (resp. terminal objects). Canonical morphism from the f ( a ) i f (a i ) f (b i ) f ( b ) i universal property of product (resp. coproduct) Unitor and counitor from adjoint a f (f (a)) f (f (b)) b relationship f f f (a) = f (f (f (a))) f (f (f (b))) = f (b) Lemma 20. Let f : F be a morphism of calyxes and F, and take elements a, a, and b, b F. Then we have the following: (1) f (a)f (a ) f (aa ) (2) f (b)f (b ) f (bb ) (3) f ((a : a )) (f (a) : f (a )) (4) f ((b : b )) (f (b) : f (b )) Notice that in one case we have equality by requirement: f (a)f (a ) = f (aa ). Proof. As stated above, (1) holds automatically. To show (2), note that f (b)f (b ) f (bb ) f (f (b)f (b )) bb And f (f (b)f (b )) = f (f (b))f (f (b ))) bb To show (3), note that and f ((a : a )) (f (a) : f (a )) f (a )f ((a : a )) f (a) f (a (a : a )) f (a) To show (4), note that f ((b : b )) (f (b) : f (b )) f (b )f ((b : b )) f (b) f (b )f ((b : b )) f (b (b : b )) f (b) Lemma 21. very ring morphism f : A B induces a calyx morphism f cal : A cal B cal where f cal is extension of ideals under f and f cal is contraction of ideals under f. cal : Rng Cal is a functor sending A to A cal and f : A B to f cal.

11 CALYXS AND COROLLAS 11 Proof. We already know there is a functor Rng Lat which comes as a restriction of the functor Grp Lat. Take rings A and B and let f : A B be a ring-homomorphism. Then f(ab) = f(a)f(b) for ideals a, b A cal. And since f(1) = 1, the extension of f(a) is B. Thus f cal is a morphism of the monoids induced by ideal product in A cal and B cal. Clearly (id A ) cal = id A cal and (g f) cal = g cal f cal for ring maps f : A B and g : B C Corollas. Definition 22. For a complete lattice M, the complete lattice = nd Lat (M) is in fact a calyx. To see this, define a map Comp : nd Lat () where φ = (φ, φ ) is sent to the adjoint pair whose left adjoint is composition by φ. xplicitly, we define Comp(φ) to send an adjoint pair ψ to the adjoint pair whose left adjoint is φ ψ and whose right adjoint is φ ψ. To see that Comp(φ) is left adjoint, take {ψ i } in. Then ) ) ( ) ( ) = φ φ ((lim (ψ i ) (x) lim ((ψ i ) (x)) = lim φ ((ψ i ) (x)) = lim (φ (ψ i ) ) since φ is left adjoint, so that Comp(φ) distributes over colimits and is left adjoint by the adjoint functor theorem for posets, lemma 2. Define Comp(φ) to be this adjoint pair. Next let {φ i } be a collection of adjoint pairs in and take ψ. Comp(lim φ i ) (ψ) = (lim φ i ) ψ = lim(φ i ψ) = lim Comp(φ i ) (ψ) It follows that Comp(lim φ i ) = lim Comp(φ i ) so that Comp(lim φ i ) = lim Comp(φ i ). Thus Comp distributes over colimits, so that by lemma 2, Comp is left adjoint. Lastly, we check associativity and unity as follows: Comp(φ ψ)(ρ) = φ ψ ρ = Comp(φ)(Comp(ψ)(ρ)) Comp(1)(φ) = 1 φ = φ Definition 23. Let be a calyx. An -corolla is a complete lattice M equipped with a calyx morphism nd Lat (M). We often write M for 1 in the lattice M, an abuse of notation. Lemma 24. An A-module M induces an A cal -corolla, called M cor. Proof. Define Φ : (A) nd Lat ((M)) by taking { n } Φ(a) (x) = ax = a i y i : a i a, y i x and i=1 Φ(a) (x) = (x : a) = {y M : ya x} Φ(a) and Φ(a) are adjoint for each a (A). Indeed, ax y x (y : a). Φ distributes over colimits, so that it is left adjoint by the adjoint functor theorem. (x) Moreover Φ(ab) (x) = abx = Φ(a) (Φ(b) (x)) and Φ(ab) (x) = (x : ab) = ((x : a) : b) = Φ(a) (Φ(b) (x))

12 12. DAN YOUNG, H. DRKSN Remark. A calyx can be viewed as a corolla over itself. By definition there is a Lat morphism nd Lat (). This morphism becomes a morphism of calyxes under the apparent calyx structure of nd Lat (). Lemma 25. The following properties hold in a general -corolla. Notice in particular that they hold for M cor for an A-module M. Reason Property Dual Property φ is a functor. a b ax bx a b (x : b) (x : a) φ (a) and φ (a) are functors. x y ax ay x y (x : a) (y : a). φ (1) = 1. 1x = x (x : 1) = x φ is left adjoint and there- 0x = 0 (x : 0) = 1 fore preserves initial objects. φ is left adjoint and therefore preserves colimits. φ (a) (resp. φ (a) is left adjoint (resp. right adjoint) and therefore preserves colimits (resp. limits). Canonical morphism from universal property of colimit (resp. limit) Unitor and counitor from adjoint relationship φ (a) φ (a) (a ix) = ( a ) i x (x : a i) = ( x : a ) i a x i = ax ( i x i : a ) = (x i : a) a x i ax i x (ax : a) a(x : a) x. (x i : a) ( x i : a ) φ (ab) = φ (a) φ (b) a(bx) = (ab)x ((x : a) : b) = (x : ab) φ (a) 1 ax x (x : a) x. φ (a) (resp. φ (a) ) is left adjoint (resp. right adjoint) and so preserves initial ob- a0 = 0 (1 : a) = 1 jects (resp. terminal objects). φ (a) φ (a) ax y x (y : a) φ (a) is a limit, φ (b) is a a(x : b) = (ax : b) right adjoint. Definition 26. Let M and N be -corollas with structure maps φ : nd Lat (M) and ψ : nd Lat (N). An -lattice morphism is a morphism f : M N in Lat such that, for each a, the following diagram commutes: M M φ(a) f f Lemma 27. very A-module morphism f : M N induces an A cal -corolla morphism f cor : M cor N cor by extension and contraction. The resulting function cor : A-mod A cal -Cor is a functor. N N ψ(a)

13 CALYXS AND COROLLAS 13 Proof. Take a M cor, b N cor. f(a) b a f 1 (b), so extension and contraction of submodules is indeed an adjoint relationship. Also extension and contraction by an identity map of modules does nothing, and (f g) cor (a) = f g(a) = f cor g cor (a) (f g) cor (a) = g 1 (f 1 (a)) = g cor f cor (a) Lemma 28. Let f : M N be a morphism of -corollas. Take elements a, {a i } in, x, {x i } in M, and y, {y i } in N. Then Reason Property Dual Property f (resp. f ) is left adjoint f ( a ) i = f (a i ) f ( b ) i = f (b i ) (resp. right adjoint) and so distributes over colimits (resp limits) f (resp. f ) is left adjoint f (0) = 0 f (1) = 1 (resp. right adjoint) and so preserves initial objects (resp. terminal objects). Canonical morphism from the f ( a ) i f (a i ) f (b i ) f ( b ) i universal property of product (resp. coproduct) Unitor and counitor from adjoint relationship f f a f (f (a)) f (f (b)) b f (a) = f (f (f (a))) f (f (f (b))) = f (b) Lemma 29. Let be a calyx and M an -corolla. Note that, for a and x, y M, ax y x (y : a). This leads to a characterization of ax. ax = lim y = lim y ax y x (y:a) Likewise, (x : a) = lim y = lim y y (x:a) ay x Lemma 30. Let f : M N be a morphism of -corollas. Then f (ax) = af (x) af (x) f (ax) f ((x : a)) (f (x) : a) f ((x : a)) (f (x) : a) Proof. The top left follows by definition. For the top right, f (ax) = f ( lim ax y y) = lim f (y) ax y lim y = f (ax) f (ax) y The others follow similarly.

14 14. DAN YOUNG, H. DRKSN 3. Universal Constructions in Lat, Cal, and -Cor In this section we calculate many of the limits and colimits in Lat, Cal, and -Cor. Lat The categry of complete lattices whose morphisms are adjoint pairs. Cal The category of calyxes and calyx morphisms. -Cor The category of -corollas with -corolla morphisms. For the case of corollas we also establish a notion of perennial morphism f : M N in -Cor, meaning that f : M N is injective on f (N) and f is injective on f (M). If f is not perennial we call it annual. Perennial morphisms more closely resemble morphisms of modules, and indeed every morphism of modules induces a perennial morphism of - corollas. The category -Cor is complete with a zero object. However, only the perennial monomorphisms are normal and only the perennial epimorphisms are conormal, so -Cor lacks the crucial property of being abelian. Restricting the category to perennial morphisms, the category loses existence of products and coproducts The Category Lat. Lemma 31. Let f : M N be a morphism in Lat. Then f (f (f (x))) = f (x) x M and f (f (f (x))) = f (x) x N. Proof. f (f (x)) x x N, so, applying f, f (f (f (x))) f (x). And f (f (y)) y y M, so taking y = f (x), f (f (f (x))) f (x). The other claim follows similarly. Theorem. Let f : M N be a morphism in Lat. The following are equivalent: (1) f (f (x)) = x x M (2) f is injective. (3) f is surjective. (4) f is a monomorphism. Proof. We show (1) (2), (1) (3), (2) (4). (1) (2). Suppose f (f (x)) = x x M. Then f (x) = f (y) x = f (f (x)) = f (f (y)) = y. (2) (1). Suppose f is injective. By 31, f (f (f (x))) = f (x) x M, so f (f (x)) = x x M. (1) (3). Suppose f (f (x)) = x x M. It follows immediately that f is surjective. (3) (1). Suppose f is surjective. By 31, f (f (f (y))) = f (y) y M. Taking x M, we can write x = f (y) for some x M, so that f (f (x)) = f (f (f (y))) = f (y) = x Clearly (2) (4) using the uniqueness of right adjoints for a given left adjoint. Suppose (2). Then take x, y M such that f (x) f (y). There are left adjoint morphisms g, h : I = {0, 1} M where g(1) = x and h(1) = y. f g = f h but g h. Theorem. Let f : M N be a morphism in Lat. The following are equivalent:

15 CALYXS AND COROLLAS 15 (1) f (f (x)) = x x N (2) f is injective. (3) f is surjective. (4) f is an epimorphism. Proof. Follows in the same way as before. xample 32. Not all monomorphisms in Lat are kernels. For example take the lattice Z Lat. There is a morphism f : I Z Lat such that f (0) = 0 and f (1) = 1. But clearly this is not a kernel. Lemma 33. Take a complete lattice Lat. The set C = {X : X is closed under sums and contains 0} forms a poset category. Note that elements X C form complete lattices with an intersection and terminal object possibly distinct from the ones in (a poset with arbitrary upper bounds and an initial object is complete). Morphisms f : X Y are lattice morphisms f : X Y such that the following diagram commutes: X f Y The category Sub() of subobjects of is categorically equivalent to C. Proof. Define a functor Φ : Sub() where a subobject f : X is sent to a = f (X). The resulting set a is closed under sums and contains the initial object 0 of the lattice, since f is left adjoint. To define an inverse map Ψ : Sub(), send a to the monomophism a whose left adjoint is the inclusion map a. Lemma 34. Take a complete lattice Lat. The set D = {X : X is closed under intersections and contains 1} forms a poset category. Note that elements X C form complete lattices with a sum and initial object possibly distinct from the ones in (a poset with arbitrary lower bounds and a terminal object is complete). Morphisms f : X Y are morphisms f : X Y such that the following diagram commutes: The category Quot() of subobjects of is categorically equivalent to D. Proof. Similar to the previous proof. X Lemma 35. There is a unique element 0 in Lat which has one element, which forms a zero object for Lat. f Definition 36. Let M be a complete lattice. We define (x) = {y M : y x} and [x] = {y M : y x}. There are canonical adjunctions i = (i, i ) : (x) M where Y

16 16. DAN YOUNG, H. DRKSN i (y) = y and i (y) = y x, and π : M [x] where π (y) = y + x and π (y) = y. To check that i i, take y (x) and z M. Then y i (z) y z x y z i (y) z To check that π π, take y M and z [x]. Then Lemma 37. Lat has kernels. y π (z) y z y + x z π (y) z Proof. Let f : M N be a morphism in Lat. Let g : (f (0)) M be the embedding and let g : M (f (0)) send x to x f (0). These form an adjoint pair: take x f (0) and y M. Then, since x f, x g (y) x y f (0) x y g = (g, g ) is a kernel for f in Lat. To show this, take a complete lattice P and a morphism h = (h, h ) : P M such that f h factors through the 0 object. Then f (h (x)) = 0 x P, so that h (x) f (f (h (x))) = f (0). Thus h factors through (f (0)) by a morphism k : P (f (0)). Define a lattice morphism k : (f (0)) P where k (x) = h (x). k and k form an adjoint pair: take x P and y f (0). Then Thus g = ker(f) in Lat. x k (y) x h (y) h (x) y k (x) y P k h g (f (0)) M N f Lemma 38. Lat has cokernels. Proof. Let f : M N be a morphism in Lat. Let g : N [f (1)] be the quotient map sending x to x + f (1) and let g : [f (1)] N be the restriction sending x to x. These form an adjoint pair: take x N and y f (1). Then, since y f (1), g (x) y x + g (0) y x y g = (g, g ) is a cokernel for f in Lat. To show this, take a complete lattice P and a morphism h = (h, h ) : N P such that h f factors through the 0 object. Then h (f (0)) = 0 x P, so that h (x) = f (f (h (x))) f (1). Thus h factors through [f (1)] by a morphism k : [f (1)] P. Define a lattice morphism k : P [f (1)] where k (x) = h (x) f (1). k and k form an adjoint pair: take x f (1) and y P. Then Thus g = cok(f) in Lat. x f (y) x h (y) h (x) y k (x) y f P M N [f (1)] h g k

17 CALYXS AND COROLLAS 17 Lemma 39. The normal monomorphisms f : M N in Lat are all equivalent (as subobjects) to (x) N for some x N. Lemma 40. The conormal morphisms f : M N in Lat are all equivalent (as quotient objects) to M [x] for some x M. Definition 41. Let {M i } be complete lattices. We form the direct sum lattice M i from the cartesian product P = {(x i ) : x i M i } where (x i ) (y i ) when x i y i i I. Take X P. It follows that x X x = ( x=(xj ) j I Xx i ) x = x X x=(x j ) j I X So that M i is complete. There are canonical morphisms ι i : M i M i. Define ι i as follows: set ι i (x) = (x j ) j I where x i = x and x j = 0 for j i. Set ι i ((x i ) ) = x i clearly ι i ι i. Lemma 42. Let {M i } be complete lattices. M i forms a coproduct of the lattices M i. Proof. Let ι i : M i M i be the canonical maps. Take a complete lattice P and morphisms f i : M i : P. To make the following diagram commute we must define f : M i P by setting f ((x i ) ) = f i(x i ). x i M i ι i M i f i f P f is then left adjoint by the adjoint functor theorem for posets, as it distributes over colimits. Lemma 43. Let {M i } be complete lattices. M i forms a product of the lattices M i. Proof. Similar to the case of coproducts. Definition 44. Let M and N be complete lattices. We form the lattice Hom Lat (M, N) from the set of adjoint pairs A = {f : f = (f, f ) : M N, f f, f : M N, f : N M} where f g when f (x) g (x) x M and f (x) g (x) x N. To show that Hom Lat (M, N) is complete it suffices to show that Hom Lat (M, N) has arbitrary joins, by the adjoint functor theorem for posets. Let {φ i } be elements of Hom Lat (M, N). Define α (x) = (φ i) (x) and α (x) = (φ i) (x). α and α are poset functors M N; since colimit is a functor, x y (φ i ) (x) (φ i ) (y)

18 18. DAN YOUNG, H. DRKSN since limit is a functor, x y (φ i ) (x) (φ i ) (y) To show α α we must show (φ i) (x) y x (φ i) (y). Observe that (φ i ) (x) y (φ i ) (x) y i I x (φ i ) (y) i I x (φ i ) (y) A 0 element for Hom Lat (M, N) is (φ, φ ) where φ (x) = 0 for each x M and φ (x) = 1 for each x N. Definition 45. Let {M i } n i=1 be lattices. Let f : i=1m i P be a set map from the n cartesian product i=1m i to a lattice P. We say f is multilinear if, for each 1 i n, and for each choice of elements X = (x 1,..., x i 1, ˆx i, x i+1,..., x n ) with x i omitted, the map f X : M i P sending x i to f (x 1,..., x n ) is left adjoint. Definition 46. Take lattices M and N, which we can view as I-corollas. We form the tensor product of lattices M I N as follows: Take M I N = ( e M N I) /, where M N is the cartesian product and is the intersection of all equivalence relations such that ( ) x i, y (x i, y) and (0, y) 0 ( x, ) y i (x, y i ) and (x, 0) 0 n x i y i i I x i y i, x i y i Notice there is a canonical map M N M I N where (m, n) is sent to the equivalence class generated by (m, n). Lemma 47 (Universal Property of Tensor Product). Let M and N be elements of Lat. Let φ : M N M I N be the canonical map. For each multilinear map f : M N P there is a unique lattice map g : M I N P such that g φ = f. Theorem. Lat forms a monoidal category under tensor product. Definition 48. Define for each set S the lattice S lat whose underlying set is the power set P(S) and such that U V in P(S) when U V. This makes ( ) lat into a functor from Set to Lat which has as a right adjoint the forgetful functor Lat Set The Category Cal. Definition 49. The unique calyx with a single element is called the 0 calyx. It is a terminal object in Cal.

19 CALYXS AND COROLLAS 19 Definition 50. We have the calyx I = {0, 1} = F cal for any field F. I is an initial object in the category Cal. Lemma 51. For an -corolla M and an element x M there is a unique -corolla morphism f x : M such that f x (1) = x. Proof. We must have f x (a) = af x (1) = ax. Defining f x in this way, we have ( ) ( ) f x a i = a i x = (a i x) = ( ) f x a i and f x (0) = 0, so that f x is indeed left adjoint. Clearly f x (ab) = af x (b). Remark. For an -corolla M and an element x M, there is not always a perennial corolla morphism f : M such that f(1) = x. We call such elements principal elements of M. For a ring A and an A module M, the regular morphisms from A cal to M cor resemble elements of M up to units. Take a vector space V over a field F. The nonzero perennial corolla morphisms from I = F cal to V cor correspond to elements of projective space over V. Definition 52. For a morphism f : F of calyxes we define ker(f) = (f (0)). It is a subcorolla, a notion to be defined later. Define im(f) = {f (a) : a } and coim(f) = {f (a) : a F }. im(f) is a subcalyx of F and coim(f) is a subcalyx of, a notion to be defined later. xample 53. Let = Z cal and F = Q cal = I. The embedding Z Q induces a morphism f : F of calyxes where f (a) = 1 for a 0, f (0) = 0, f (0) = 0, f (1) = 1. ker(f) = {0}. im(f) = I, coim(f) = {0, 1}. ven though ker(f) = 0, f is not injective, and the obstruction to this is the difference between and coim(f). Lemma 54. Let f : F be a calyx morphism. f is a monomorphism in Cal if and only if it is a monomorphism in Lat. Proof. If f is a monomorphism in Lat then a priori it is a monomorphism in Cal. Conversely, by lemma 3.1 it suffices to show that if f : F is not injective then f is not a monomorphism in Cal. So take a, b in such that f (a) = f (b). Then by 51 there are morphisms g : and h : such that g(1) = a and h(1) = b. Then f g = f h but g h. Lemma 55. An epimorphism in Lat is an epimorphism in Cal Proof. Follows from 3.1 Definition 56. Let be a calyx. A subcalyx of is a set F closed under sums and products and contianing 0. For each x, (x) = {y : y x} forms a subcalyx, called the principal subcalyx with respect to x. It follows that (x), or any subcalyx of, forms a calyx in its own right, as an adjoint pair is determined by a left adjoint. The quotient and intersection in a subcalyx of is distinct from the quotient and intersection in. Lemma 57. Take a calyx. The category Sub() of subobjects of is equivalent to the category C of subcalyxes X of whose morphisms f : X Y are calyx epimorphisms f : X Y making the following diagram commute:

20 20. DAN YOUNG, H. DRKSN X f Y Proof. Define a functor Φ : Sub() C where a subobject f : X is sent to im(f). For an inverse functor define Ψ : C Sub() where a subcalyx X is sent to the morphism f : X whose left adjoint is the inclusion poset functor X. Definition 58. Let be a calyx. A quotient calyx of is a set F closed under intersections and ideal quotients, and containing 1. For each x, [x] = {y : y x} forms a quotient calyx, called the principal quotient calyx with respect to x. It follows that [x], or any subcalyx of, forms a calyx in its own right, as an adjoint pair is determined by a right adjoint. The product and sum in a quotient calyx of is distinct from the product and sum in. Lemma 59. Take a calyx. The category Quot() of quotient objects of is equivalent to the category D of quotient calyxes of whose morphisms f : X Y are calyx epimorphisms f : X Y making the following diagram commute: X f Proof. Define a functor Φ : Quot() D where a quotient object f : X is sent to coim(f). For an inverse functor define Ψ : D Quot() where a quotient object X of is sent to the morphism f : X whose right adjoint is the inclusion poset functor X. Theorem. Cal forms a complete category. Definition 60 (Tensor product of calyxes). Definition 61. There is a functor Φ : Lat Cal which sends a complete lattice to the calyx i=0 n where 3.3. The Category -Cor Morphisms in -Cor. (a 1 a n )(a n+1 a n+m ) = a 1 a n+m Definition 62. Let f : M N be a (possibly annual) morphism of -corollas. We define the following sets, called the kernel, cokernel, image, and coimage respectively: Y ker(f) = (f (0)) cok(f) = [f (1)] im(f) = (f (1)) coim(f) = [f (0)] We will later see that ker(f) and im(f) are subcorollas, while cok(f) and coim(f) are quotient corollas.

21 CALYXS AND COROLLAS 21 Definition 63. Let f : M N be a morphism of -corollas. The following are equivalent: (1) im(f) = (f (1)) (2) f is injective on (f (1)) If these hold we say f is perennial. Definition 64. Let f : M N be a morphism of -corollas. The following are equivalent: (1) coim(f) = [f (0)] (2) f is injective on [f (0)] If these hold we say f is perennial. Definition 65. If f and f are perennial we say f is perennial. Lemma 66. An A cal -corolla morphism induced by an A-module morphism is perennial. Proof. Lemma 67. Let f : M N be a perennial -corolla morphism. Then ker(f) = {x M : f (x) = 0} cok(f) = {y N : f (y) = 1} im(f) = {f (x) : x M} coim(f) = {f (y) : y N} Proof The Category -Cor. Lemma 68. Let f : M N be a perennial morphism of -corollas. ker(f) is the obstruction to the injectivity of f (equivalently, the surjectivity of f ) and cok(f) is the obstruction to the surjectivity of f (equivalently, the injectivity of f ). More precisely, the following are equivalent: (1) f f = id (2) f is a monomorphism. (3) f is injective. (4) f is surjective. (5) ker(f) = 0 (6) coim(f) = 1 Proof. Obvious. Lemma 69. Analogously, the following are equivalent for a perennial morphism f : M N of -corollas: and the following are equivalent: (1) f f = id (2) f is an epimorphism. (3) f is surjective. (4) f is injective. (5) im(f) = 1 (6) cok(f) = 0 Proof. Obvious.

22 22. DAN YOUNG, H. DRKSN Definition 70. Let M be an -corolla. For an element x M let (x) = {y M : y x} and let [x] = {y M : y x}. We show that (x) and [x] are -corollas in their own right. Lemma 71. Let M be an -corolla with element x M. (x) can be made into an - corolla in its own right. It is already a complete lattice. Define φ : nd Lat ((x)) where φ(a) (y) = ay, where ay is multiplication in the corolla M. This determines an adjoint map φ(a) : (x) (x) which is possibly distinct from the quotient operation in M. We have φ(a) (y) = sup z = sup z z (x),z φ(a) (y) z (x),az y Lemma 72. Let M be an -corolla with element x M. The quotient (y : a) in (x) is equal to (y : a) x where is multiplication in M. Proof. By the uniqueness of adjoints it suffices to show that and y ya is left adjoint to y (y : a) x in [x]. Lemma 73. Let M be an -corolla with element x M. [x] can be made into an - corolla in its own right. It is already a complete lattice. Define φ : nd Lat ([x]) where φ(a) (y) = (y : a), where (y : a) is the quotient operation in M. This determines an adjoint map φ(a) : (x) (x) which is possibly distinct from the quotient operation in M. We have φ(a) (y) = inf z = inf z z [x],φ(a) (y) z z [x],y (z:a) Lemma 74. Let M be an -corolla with element x M. The multiplication ay in [x] is equal to a y + x where is multiplication in M Proof. By the uniqueness of adjoints it suffices to show that y a y + x is left adjoint to y (y : a) in [x]. Lemma 75. Let M be an -corolla and take x M. There is a canonical -corolla morphism (x) M and a canonical -corolla morphism M [x]. Proof. Take the canonical map i : (x) M in Lat and π : M [x] in Lat and note that i(ay) = ay = ai(y) and π(ay) = ay + x = a(y + x) + x = ay + ax + x = a(π(y)) Definition 76. We have the zero corolla 0, the unique -corolla with one element. Proof. Take an -corolla M. It suffices to note that the unique lattice morphisms 0 M and M 0 are in fact corolla moprhisms. Lemma 77. Let f : M N be an -corolla morphism. Then the canonical map g : ker(f) M is a kernel for f in the categorical sense. Lemma 78. Let f : M N be an -corolla morphism. Then the canonical map g : N cok(f) is a cokernel for f in the categorical sense. Lemma 79. Let f : M N be an -corolla morphism. Then the canonical map g : M im(f) is an image for f in the categorical sense. Lemma 80. Let f : M N be an -corolla morphism. Then the canonical map g : coim(f) N is a coimage for f in the categorical sense.

23 CALYXS AND COROLLAS 23 Lemma 81. Let f : M N be a perennial morphism of -corollas. f is normal if and only if f is a monomorphism. Lemma 82. Let f : M N be a perennial morphism of -corollas. f is conormal if and only if f is an epimorphism. Definition 83. Let {M i } be -corollas. We form the corolla M i from the cartesian product P = {(x i ) : x i M i } where (x i ) (y i ) when x i y i i I. Take X P. It follows that x X x = ( ) x=(xj ) j I Xx i x = Take x = (x i ) P. We define x X x=(x j ) j I X ax = (ax i ) (x : a) = ((x i : a)) It is routine to check that this satisfies the requirements of an -corolla. There are canonical monomorphisms ι i : M i M i where x i maps to the element (x i ) where x j = 0 for j i. Likewise, there are canonical morphisms π i : M i M i where (x i ) maps to x i. Lemma 84. Let {M i } be -corollas. M i forms a coproduct for {M i } Proof. Let P be an -corolla and take -corolla morphisms f i : M i P. We must define f : M i P such that f ι i = f i. Take an element (x i ) in M i and let X i be the element whose ith entry is x i and which is 0 elsewhere. Then we must have ( ) f ((x i ) ) = f X i = f (X i ) = f (ι i (x i )) = f i (x i ) Defining f in this way produces a poset functor which distributes over coproducts and preserves the initial object 0, so that it is left adjoint. Moreover f (a(x i ) ) = af ((x i ) ), so that the lattice morphism f induced by f is an -corolla morphism, as claimed. Lemma 85. Let {M i } be -corollas. M i forms a product for {M i } Proof. Let P be an -corolla and take -corolla morphisms f i : P M i. We must define f : P M i such that π i f = f i. Take an element (x i ) in M i and let X i be the element whose ith entry is x i and which is 1 elsewhere. Then we must have ( ) f ((x i ) ) = f X i = f (X i ) = f (π i (x i )) = f i (x i ) Defining f in this way produces a poset functor which distributes over products and preserves the terminal object 1, so that it is right adjoint. Moreover f (ax) = f i (ax) = ( ) af ii (x) = a f i (x) = af (x) so that the lattice morphism f induced by f is an -corolla morphism, as claimed. Theorem. -Cor forms a complete category. x i

24 24. DAN YOUNG, H. DRKSN Definition 86. Let M and N be -corollas. We form the corolla Hom (M, N) as the set of -corolla morphisms f : M N where f g when f (x) g (x) x M and f (x) g (x) x N. To show that Hom (M, N) is complete it suffices to show that Hom (M, N) has arbitrary joins, by the adjoint functor theorem for posets. Let {φ i } be elements of Hom (M, N). Define α (x) = (φ i) (x) and α (x) = (φ i) (x). α and α are poset functors M N; since colimit is a functor, x y (φ i ) (x) (φ i ) (y) since limit is a functor, x y (φ i ) (x) (φ i ) (y) Moreover, (φ i ) (ax) = a(φ i ) (x) = a (φ i ) (x) and i ) (φ ((x : a)) = ( ) ((φ i ) (x) : a) = (φ i ) (x) : a To show α α we must show (φ i) (x) y x (φ i) (y). Observe that (φ i ) (x) y (φ i ) (x) y i I x (φ i ) (y) i I x i ) (φ (y) A 0 element for Hom (M, N) is (φ, φ ) where φ (x) = 0 for each x M and φ (x) = 1 for each x N. Lemma 87. For a commutative calyx, Hom (, M) = M as -corollas. Proof. Define a lattice morphism Φ : M Hom (, M) where x M is sent to the morphism f x Hom (, M) where a ax. Then Φ ( x i) = Φ (x i ) and Φ (ab) = aφ (b). Φ is surjective and injective, so Φ = (Φ, Φ ) is an isomorphism, where Φ is the right adjoint of Φ. Definition 88. Let {M i } n i=1 be -corollas. Let f : i=1m i P be a set map from the n cartesian product i=1m i to a lattice P. We say f is multilinear if, for each 1 i n, and for each choice of elements X = (x 1,..., x i 1, ˆx i, x i+1,..., x n ) with x i omitted, the map f X : M i P sending x i to f (x 1,..., x n ) is a left adjoint morphism of -corollas. n

25 CALYXS AND COROLLAS 25 Definition 89. Take -corollas M and N. We form the tensor product of corollas M N as follows: Take M N = ( e M N ) /, where M N is the cartesian product and is the intersection of all equivalence relations such that ( ) x i, y (x i, y), (ax, y) a(x, y), and (0, y) 0 ( x, ) y i (x, y i ), (x, ay) a(x, y), and (x, 0) 0 x i y i i I x i y i, x i y i x y ax ay Notice there is a canonical multilinear map M N M I N where (m, n) is sent to the equivalence class generated by (m, n). Lemma 90 (Universal Property of Tensor Product). Let M and N be elements of -Cor. Let φ : M N M I N be the canonical multilinear map. For each multilinear map f : M N P there is a unique morphism of -corollas g : M I N P such that g φ = f. Lemma 91. The functor M M N is left adjoint to the functor M Hom (M, N). Theorem. -Cor forms a monoidal category under tensor product. Remark. Lat is categorically equivalent to I-Cor. As such, Lat can be viewed as a monoidal category. 4. Commutative Calyxes and their Corollas Definition 92. A calyx is called commutative if ab = ba a, b. In this section, all calyxes are assumed to be commutative Maximal, Minimal, Irreducible, Co-Irreducible, and Prime lements [-1]. Definition 93. Let be a calyx with element p. we make the following two symmetrical definitions: (1) p is called irreducible if a i p a i p for some i I. (2) p is called co-irreducible if p a i p a i for some i I. Notice that 0 is irreducible when a i = 0 a i = 0 for some i I and that 1 is coirreducible when = 1 a i = 1 for some i I. Notice that p is irreducible in if and only if 0 (the smallest element in [x], i.e. x) is irreducible in [x] (the quotient calyx) and p is co-irreducible if and only if 1 (the largest element in (x), i.e. x) is co-irreducible in (x) (the subcalyx). xample 94. In the calyx I = {0, 1}, 0 is irreducible and co-irreducible and 1 is irreducible and co-irreducible. Definition 95. Let be a calyx with element x. We make the following two symmetrical definitions: (1) x is called maximal if x < 1 and x < y y = 1. quivalently, if [x] = I.

26 26. DAN YOUNG, H. DRKSN (2) x is called minimal if x > 0 and x > y y = 0. quivalently, if (x) = I. Lemma 96. Notice that maximal elements are irreducible and minimal elements are coirreducible, since [x] = I for a maximal element and (x) = I for a minimal element. Thus, if an element m is maximal, then m x y x or z x Lemma 97. Suppose is a calyx. Then for every element a there is a maximal element m such that a m and a minimal element ζ such that ζ a. This follows from Zorn s lemma. We can form the intersection of all maximal elements containing an element and the sum of all minimal elements contained in an element. Lemma 98. Suppose a has a + m = 1 for each maximal element m. Then a = 1. Proof. By contrapositive: if a 1 then a is contained in some maximal element m, so that a + m = m 1. Lemma 99. Suppose a has a ζ = 0 for each minimal element ζ. Then a = 0 Proof. By contrapositive: if a 0 then a contains some minimal element ζ, so that a ζ = ζ 0. We next establish the notions of prime elements. Definition 100. Let be a calyx. We say an element p is prime if ab p a p or b p. quivalently, if we have ab = 0 a = 0 or b = 0 in [p]. Lemma 101. Let be a calyx. If p is prime then p is irreducible. Proof. If a b p then ab a b p, so that a p or b p. Lemma 102. Let S {0} be closed under multiplication. Suppose that x, y, x y, x S y S. Then there is a prime ideal p not in S. Proof. The set S is nonempty since it contains 0, and closed under suprema of tosets. By Zorn s lemma, there is a maximal ideal p among those not contained in S. Take a, b p such that ab p. Since a, b p, a, b S, so ab S. Thus p ab, so that p S. Lemma 103. Let be a calyx. A maximal element m is prime. Proof. m is maximal /m = I (ab = 0 a = 0 or b = 0) in /m m is prime A priori a maximal element m is irreducible. Thus we have the following diagram of implications in any calyx: Minimal Co-Irreducible Maximal Prime Irreducible Definition 104. We say that x M is irreducible if y z x y x or z x. We say x M is coirreducible if x y + z x y or x z.