The Essentially Equational Theory of Horn Classes

The Essentially Equational Theory of Horn Classes Hans E. Porst Dedicated to Professor Dr. Dieter Pumplün on the occasion of his retirement Abstract It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic [2]. In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known [1]. Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes are in fact equivalent to quasivarieties. Mathematics Subject Classification 1991. Primary 03C07, Secondary 08C15. Key words: Universal Horn class, quasivariety, essentially equational theory. Introduction The class of locally finitely presentable categories is known to comprise all complete categories of finitary many-sorted algebras see [3]. There is a universal syntax by which locally finitely presentable categories can be specified, namely that of an essentially equational theory, which generalizes the classical syntax of equational classes of algebras to a specification of certain classes of partial algebras see [2]. While it is easy to specify implicationally defined classes of algebras this way see [1], a canonical specification for Horn classes which are known to be locally finitely presentable categories by an essentially equational theory seems not to be known: The univeral representation presented in [2] might be viewed unnatural: e.g., representing POS the category of partially ordered sets and monotone maps this way would require ℵ 0 sorts, whereas a 1 sorted representation is possible, as is shown in [5]. We are to provide in this note combining ideas of [1] and [5] an essentially equational representation of Horn classes much simpler than in [2], which moreover in many cases represents Horn classes as quasivarieties. The main idea is to introduce, for each relational symbol provided by the signature under consideration, a new sort 1

and a family of operations corresponding to the respective projections; this enables us, in a second step, to represent relational implications in terms of partial operations and equations. For a related approach see [4]. Preliminaries Language and notation are mainly as in [2]. To make this note as selfcontained as possible we briefly explain the basic notions. Products of finite families of sets or maps are denoted by n A i or A 1... A n respectively n f i or f 1... f n according to convenience; the product projection onto the i-th component of a subset of a product is denoted by π i, with superscripts referring to the particular product if necessary. The map B n A i induced by a family of maps g i : B A i...n, i.e., the map g with π i g = g i for each i is denoted by g i. Let Σ = Σ op Σ rel be an S sorted finitary signature, i.e., S is a set of sorts, Σ op is set of operational symbols, Σ rel is a set of relational symbols, and for each symbol ω Σ there is given an arity arω = s 1,..., s n, s 0 S n+1 for some n N. In view of the role the arities are to play we will use more suggestive notions for them as follows: for σ Σ op we write s 1... s n s 0 instead of s 1,..., s n, s 0. for σ Σ rel we write s 1... s n s 0 instead of s 1,..., s n, s 0. A Σ-structure then is a triple A = A s s S, σ A σ Σop, R A R Σrel where A s s S is a family of S-indexed sets, σ A : A s1... A sn A s0 is a map for each σ Σ op with arσ = s 1... s n s 0, and R A A s1... A sn A s0 is a subset for each R Σ rel with arr = s 1... s n s 0. A morphism of Σ-structures f : A B is an S-indexed family of maps f s : A s B s s S satisfying the conditions σ B f s1... f sn = f s0 σ A for each σ Σ op with arσ = s 1... s n s 0, f s1... f sn [R A ] R B for each R Σ rel with arr = s 1... s n s 0. This defines the category StrΣ of Σ-structures which by means of the obvious underlying functor StrΣ Set S is a concrete category over Set S, the S-fold power of the category of sets and mappings. The signature Σ is called finite provided both, S and Σ, are finite sets. As usual Σ is called a signature of algebras if Σ rel is empty; in this case Σ-structures are simply denoted as pairs A s s S, σ A σ Σop. 2

The following particular types of subcategories of categories of the form StrΣ are of special interest: Horn classes: Let there be given a set I of Σ implications α 1 α 2... α n = β where n N and α i and β are atomic Σ formulas i.e., either Σ op -equations, or formulas Rt 1,..., t m where R Σ rel is an m ary relational symbol and t 1,..., t m are terms of the corresponding sorts. The full subcategory ModΣ, I of StrΣ spanned by all Σ structures satisfying the implications in I is then called a universal Horn class. If, for each implication in I, the conclusion β is in fact an equation, we call the Horn class quasivarietal. In case Σ rel = the category ModΣ, I is called a quasivariety. The category POS of partially ordered sets is an example of a Horn class, which is not a quasivariety. Horn classes are locally finitely presentable categories see [2]. Categories of partial algebras: Denote by PAlgΣ the full subcategory of StrΣ consisting of those Σ-structures A for which each relation R A R Σ rel, arr = s 1,..., s n, s 0 satisfies a 1,..., a n, a 0, a 1,..., a n, b 0 R A = a 0 = b 0. In other words, for any A in PAlgΣ each relation R A with arity s 1,..., s n, s 0 is a partial map from some subset domr A A s1... A sn into A s0. Considering categories of the form PAlgΣ we might speak of partial operational symbols instead of relational symbols and denote their arities the same way as for total operational symbols; we also might write Σ p instead of Σ rel and Σ t instead of Σ op in this case, and then call Σ a signature of partial algebras. Essentially equational essentially algebraic categories: By an essentially equational theory Ω, def, E with set of sorts S we mean a S sorted finitary signature Ω = Ω t Ω p of partial algebras together with finite sets defω of Ω t equations for each ω Ω p the domain conditions of ω, where for arω = s 1... s n s 0 the elements of defω are equations in the S-sorted variables x s1,..., x sn, and a set E of Ω equations the identities of the theory. The theory is called finite if S and Ω are finite sets. A partial Ω algebra A is called a model of this theory iff i the operations ω A are total for each ω Ω t ; ii for ω Ω p with arω = s 1... s n s 0 and a 1,..., a n A s1... A sn one has a 1,..., a n domω A iff t A a 1,..., a n = t A a 1,..., a n for all equations t t defω. iii A satisfies the identities of E 1. 1 Satisfaction here is meant in the weak sense: if both sides of the equation are defined then they are equal. 3

These models form the full subcategory ModΩ, def, E of PAlgΩ which thus is a concrete category over Set T. Categories of the form ModΩ, def, E for essentially equational theories Ω, def, E are called essentially algebraic or essentially equational categories. It is shown in [2] that the locally finitely presentable categories are up to equivalence of categories precisely the essentially equational categories. A paradigmatic example of a 1-sorted essentially equational category ist the category Cat of small categories and functors: the total operations are besides the identities as constants domain and codomain, the only partial binary operation is composition and defcomposition consists of the single equation domain x = codomain y. Cat is not equivalent to a Horn class. Each quasivariety has a canonical representation by an essentially equational theory as follows see [1]: add to the given siganture, for each implication ϱ 1 τ 1... ϱ n τ n = π τ, one partial operation ω with domain condition defω = {ϱ i τ i i = 1,..., n} together with the equations ω π and ω τ. Results For a given S sorted finitary signature Σ = Σ op Σ rel we define an S sorted finitary signature of algebras Ω Σ as follows: S = S {s R R Σ rel } is the set of sorts. Ω Σ op = Σ op {Σ R R Σ rel } is its set of operational symbols, where, for R Σ rel of arity s 1... s n, the set Σ R consists of n = nr unary operational symbols R π i i = 1,..., n with arities s R s i ; the arity of σ Σ op is the one given by the signature Σ. Note that Ω Σ is finite provided Σ is finite. Proposition 1 For every S sorted finitary signature Σ there exist a finitary S sorted signature of algebras Ω Σ, a set I Σ of Ω Σ implications, and functors StrΣ GΣ ModΩ Σ, I Σ F Σ Str Σ such that F Σ G Σ = 1 and G Σ F Σ 1. If the signature Σ is finite, so is the signature Ω Σ and also the set of implications I Σ. The above proposition being only a slight generalization of [2, 3.20] is included here, as well as its proof, because it serves as a basis for the proofs of the following results. 4

Proposition 2 Let Σ be an S sorted finitary signature and ModΣ, J StrΣ a quasivarietal Horn class. Then there exists a set I J of Ω Σ implications of the same cardinality as J such that G Σ and F Σ can be restricted to an equivalence between ModΣ, J and the S sorted quasivariety ModΩ Σ, I Σ I J. Proposition 3 Let Σ be an S sorted finitary signature and H = ModΣ, J StrΣ an arbitrary universal Horn class. Then there exists an essentially equational theory Ω H, def, E H with set of sorts S such that H and ModΩ H, def, E H are equivalent. If Σ is a finite signature and the set J of implications is finite, then Ω H, def, E H is a finite theory. Proofs Proof of Proposition 1. For Σ = Σ op Σ rel with set of sorts S we have S = S Σ rel and Ω Σ = Σ op {Σ R R Σ rel }. Define I Σ as { } I Σ = R π 1 x R π 1 y... R π n x R π n y = x y R Σ rel Given a Σ structure A = A s s S, σ A σ Σop, R A R Σrel define G Σ A = Āt t S, σā σ Σop, R πāi R πi ΣR, R Σrel by { As if t = s S Āt : = R A if t = s R, R Σ rel σā : = σ A R πāi : = nr A si R A π i A si i.e., R πāi is the restricted product projection. G Σ A then obviously satisfies all implications in I Σ. G Σ becomes a functor if one defines, for a StrΣ morphism h = h s s S : A B, G Σ h = ht : Ā t B t by t S h s : A s B s if t = s S nr h t = h si : R A R B 2 if t = s R, R Σ rel 2 here, clearly, h sr is meant to be h si restricted and corestricted to R A and R B respectively. 5

Now define, for any Ω Σ, I Σ model A = A t t S, σ A σ Ω Σ op, the Σ model by F Σ A = Ãs s S, σã σ Σop, RÃ R Σrel Ãs : = A s, σã : = σ A, RÃ : = im R π A i i.e., RÃ is the image of the map R π A i : A sr F Σ becomes a functor if we define F Σ h A t h t Bt t S = s As Bs s S. It is obvious that the equation F Σ G Σ = 1 holds. Also for A = A t t S, σ A σ Ω Σ and G Σ F Σ A = Ã- t t S, σ Ã σ Ω Σ one obviously has Ã- s = A s for each s S σ Ã = σ A for each σ Σ op For R Σ rel we have Ã- s R = RÃ = im R πi i A. Since A satisfies the implications of I Σ, the map R πi A i is monic, hence a bijection χ A R : A sr RÃ. Similarly, for h = h t t S : A B in ModΩ Σ, I Σ and G Σ F Σ h = ht S one has hs = h s for each s S. For R Σ rel the map hsr : RÃ R B is the co/restriction of nr h si. Hence, in the following diagram all cells commute. nr A si. A sr h sr B sr χ A R RÃ h R R B χ B R R π A i Asi Q hsi Bsi R π B i π i π i A si hsi B si 6

Thus, the S sorted maps χ A t : A t à t with { χ A 1As if t = s S t = χ A R if t = s R, R Σ rel are isomorphisms A G Σ F Σ A, natural in A. Lemma Let Σ be a S sorted finitary signature and α = Qt 1,... t n an atomic Σ formula with Q a relational symbol of arity s 1... s n. Let E α be the set of Ω Σ equations E α = { Q π 1 t 1,..., Q π n t n }. Then the following holds: i If ā = a 1,..., a m is an assignment to variables in the Σ structure A such that t A 1 ā,..., t A n ā Q A, then there exists an assignment ā = b, a 1,..., a m in the Ω Σ, I Σ model G Σ A fulfilling all equations in E α. ii Let b = b, a 1,..., a m be an assignment to variables in the Ω Σ, I Σ model A such that b fulfills all equations in E α ; then t F Σ A 1 b,..., t F Σ A n b Q F Σ A for the assignment b = a 1,..., a m in the in the Σ structure F Σ A. Proof x 1,..., x m denote the S sorted variables needed by t 1,..., t n, x i of sort s i. Let A = A s, σ A σ, R A R be a Σ structure and ā = a1,..., a m with a i A si an assignment to the variables x 1,..., x m, such that t A i ā,..., t A n ā Q A. Then ā = b, a 1,..., a n with b = t 1 ā,..., t n ā is an assignment to the S sorted variables x, x 1,..., x m with x of sort s Q in the Ω Σ model G Σ A. To prove that ā fulfills all equations in E α now is trivial. Conversely, if b = b, a 1,..., a m is an assignment to the S sorted variables x, x 1,..., x m with x of sort s Q in the ModΩ Σ, I Σ model A = A t t, σ A σ fulfilling all equations in E α a straightforward calculation shows that for the assignment b = a 1,..., a m to the S sorted variables x 1,..., x m in the Σ structure F Σ A one has t F Σ A 1 b,..., t F Σ A n b Q F ΣA. Proof of Proposition 2. For each implication J : α 1... α k = s t we consider the sets E αi of Ω Σ equations as in the Lemma where applicable. If α i = s i t i is an equation we define E αi to be {s i t i }. By e i we denote the conjunction of the equations of E αi, i.e., { Ri π e i = 1 t i 1... Ri π ni t i n i if αi = R i t i 1,... t i n i s i t i if α i = s i t i 7

Let now J be the implication J : e 1 e 2... e k = s t and I J the set of all implications J for J J. It remains to show that G Σ maps H = ModΣ, J into ModΩ Σ, I Σ I J and that F Σ maps this category back into H. Let A = A s s, σ A σ, R A R be in H. It suffices to show that G Σ A satisfies all implications of the form J. Let now b be an assignment to the variables needed for J in G Σ A satisfying the premis J. Then b satisfies all equations in E αi for each i = 1,..., k. By ii of the Lemma there results an assignment b in F Σ G Σ A = A such that each atomic formula α i is satisfied by b. Since J holds in A we conclude s A b = t A b, hence GΣA s b = GΣA t b, since b and b differ only in not assigning values to variables of the sorts s Ri which don t occur in s and t respectively. Conversely, let A = A t t, σ A σ be in ModΩ Σ, I Σ I J and ā an assignment to the variables needed in J J in the Σ structure F Σ A, such that each atomic formula α i occuring in the premis of J is satisfied by ā. By i of the Lemma there exists an assignment ā in G Σ F Σ A fulfilling the equations in E αi for all i. Since G Σ F Σ A and A are isomorphic by Proposition 1 and A satisfies the implications in I J so does G Σ F Σ A; hence we conclude s GΣ F ΣA ā = t GΣ F ΣA ā, thus s F ΣA ā = t F ΣA ā. Before considering the general case of an arbitrary Horn class H StrΣ we recall from the introduction the description of the quasivariety ModΩ Σ, I Σ by means of an essentially equational theory: Ω Σ, def, E Σ is the essentially equational theory with S as set of sorts, Ω Σ t = Ω Σ op Ω Σ p = { R R Σ rel }, where for each R Σ rel of arity s 1... s n the corresponding R is considered as a binary operational symbol of arity s R s R s R with domain condition def R = { R π i x R π i y i = 1,..., n }. E Σ = { } E R with E R = Rx, y x, Rx, y y R Σ rel It is well known and easy to see that the obvious forgetful functor yields an isomorphism Mod Ω Σ, def, E Σ ModΩ Σ Mod Ω Σ, def, E Σ ModΩ Σ, I Σ. 8

Proof of Proposition 3. We continue as in the preceding remark and enlarge the theory Ω Σ, def, E Σ to Ω H, def, E H as follows: Ω H t : = Ω Σ t Ω H p : = Ω Σ p = Ω Σ op { J J J } = Σ rel { J J J } where for each implication J : α 1... α k = β in J with α i = R i t i 1,..., t i n i, β = Qt 1,..., t m and needed variables x 1,..., x l of sorts s 1,..., s l S J is a partial operational symbol with arity with domain condition def J = k E αi, E H = E Σ E J with E J : = s R1... s Rk s 1... s l s Q 3 J J Observe first that the obvious forgetful functor { } Q π i J ti i = 1,..., m. V : ModΩ H, def, E H Mod Ω Σ, def, E Σ is full: if h = h t t : V A V B is a homomorphism in Mod Ω Σ, E Σ and J J it is to be shown that, for each ā = b 1,... b k, a 1,..., a l fulfilling all equations in def J in the algebra A, hā fulfills these equations in B and also the homomorphism condition h Q J A ā = J B h sr1... h srk h s1... h sl ā = J B hā. To prove this, consider ā = b 1,..., b k, a 1,..., a l dom J A with b i of sort s Ri, a j of sort s j. ā dom J A implies b i = R i πj A i ā = t i j i A ā for all i, j i ; since h is a morphism in ModΩ Σ, I Σ one has h sri t i j i A ā = t i j i B hā for all i, j i, and therefore R i π B j i hā = R i π B j i hsri b i = t i j i B hā, thus, hā dom J B. Now, for ā dom J A, one concludes for all terms t i of sorts s i occuring in the conclusion β: Q πi B J B hā = t B i hā = h s i t A i ā = h s Q i πi A J A ā = Q πi B h Q J A ā 3 omit here s Ri if α i is of the form t 1 t 2 9

Since V B satisfies the implications in I Σ, h is a homomorphism. Next, we lift G Σ and F Σ along V to obtain functors G and F such that the diagram ModΣ, J G ModΩ H, def, E H F ModΣ, J StrΣ G Σ ModΩ Σ, I Σ V F Σ StrΣ commutes, which immediately yields F G = 1 and, since V is full and faithful, also G F 1. Let A = A s s, σ A σ, R A R satisfy all implications J J. Consider G Σ A = Āt, σ A σ Ω Σ t, R A R Σrel we identify, for the sake of simplicity, Mod Ω Σ, def, E Σ and ModΩ Σ, I Σ. Define GA: = Āt, σ A σ Ω Σ t, R A R Σrel, J A J J, where for J : R 1 t 1 1,..., t 1 n 1... R k t k 1,..., t k n k = Qt 1,..., t m * in J with S sorted variables x 1,... x l of sorts s 1,... s l needed for the t j j i, t i, the partial operation with R A 1... R A k l A si dom J A dom J A = {b 1,..., b k, ā b j R A j, ā is defined by Then any b 1,..., b k, ā k Rj A l J A Q A l A si, b j = t j 1ā,..., t j n j ā } J A b 1,..., b k, ā = t 1 ā,..., t m ā j=1 A si with b j = b j 1,..., b j n j satisfies all equations in def J iff, for all j, i j, the equation b j i j = R j π A i j b j = t j ij A ā holds, i.e., iff b 1,..., b k ā dom J A, and, if so, J A b 1,..., b k, ā Q A since A satisfies the implication J. GA satisfies the identities of E J, hence of E H, since for each b 1,..., b k, ā as above one has J A b 1,..., b k, ā = Q π i t A 1 ā,..., t A mā Q π A i = t A i ā = t A i b 1,..., b k, ā. 10

Since V is full we can define Gh = G Σ h for morphisms h in ModΣ, J. In order to define F we only need to show that, for each A in ModΩ H, def, E H, A = A t t, σ A σ, R A R, J A J, the Σ-structure F Σ V A = Ãs s, σã σ, Rà R satisfies all implications J J. Hence let J J be as in * and ā an assignment to the variables x 1,..., x l in the algebra F Σ V A such that for j = 1,..., k bj : = t j 1Ãā,..., t j n j Ãā RÃj = im R j π A i. Then there exist uniquely determined elements recall that the maps R j πi A are monic b j Rj A j = 1,..., k with R j π A i j b j = t j i j A ā for all j, i j. Since A is an Ω H, def, E H model this means in particular: b 1,..., b k, ā dom J A, J A b 1,..., b k, ā Q A, and Q πi A J A b 1,..., b k, ā = t A i ā for i = 1,..., m. It follows t à 1 ā,..., tãmā = t A 1 ā,..., t A mā = Q πi A J A b 1,..., b k, ā QÃ. References [1] J. Adámek, H. Herrlich, and J. Rosický. Essentially equational categories. Cahiers Topol. Géom. Différentielles Catégoriques, 24:175 192. [2] J. Adámek and J. Rosicky. Locally presentable and accessible categories. Cambridge, 1994. [3] M. Makkai and R. Paré. Accessible Categories. Contemp. Math., volume 104. Amer. Math. Soc., Providence R.I., 1989. [4] T. Mossakowski. Equivalences among various logical framework of partial algebras. In Lecture Notes in Computer Science, volume 1092, pages 403 433, 1996. [5] H. -E. Porst. The algebraic theory of order. Appl. Categorical Structures, 1:423 440, 1993. Department of Mathematics, University of Bremen, 28359 Bremen, Germany porst@math.uni-bremen.de 11