On injective constructions of S-semigroups Jan Paseka Masaryk University Joint work with Xia Zhang South China Normal University BLAST 2018 University of Denver, Denver, USA Jan Paseka (MU) 10. 8. 2018 0 / 39
Contents 1 Introduction M Motivation S Sierpiński space and injectivity B Complete Boolean algebras and injectivity Q Backgrounds: Quantales and quantale-like structures H Backgrounds: Injective hulls for some partially ordered algebras 2 Injective hulls of posemigroups O Injective objects of posemigroups C Construction of a special closure operator H Injective hulls of posemigroups 3 Injective hulls for S-semigroups O Injective objects of S-semigroups C Construction of a special closure operator H Injective hulls of S-semigroups 4 References Jan Paseka (MU) 10. 8. 2018 1 / 39
Introduction: Motivation Fuzzy logic It is well-known that the semantics of a given fuzzy logic can be formally axiomatized by means of a residuated poset. The appropriateness of such approach is emphasized by the fact that t-norm based logic usually refers to residuated systems of fuzzy logic with t-norm based semantics. Here the conjunction connective is interpreted by a t-norm and the implication operator by its residuum. In essence, this fact is the source of numerous examples of prospective truth functions of connectives for fuzzy logic. Hence, we assume that a corresponding residuated poset is given and its connection to the semantics of a fuzzy logic is known. Content of the work In this paper, we shall investigate the injective hulls in the category of S-semigroups over a posemigroup S and we obtain their concrete form. Jan Paseka (MU) 10. 8. 2018 2 / 39
1. Introduction: Sierpiński space and injectivity We shall write S for the Sierpinski space, i.e., the set 2 = {0, 1} equipped with topology Ω(S) = {S, {1}, }. Statement (i) Any T 0 -space may be embedded as a subspace of a power of S. (ii) S is an injective T 0 -space, i.e., if i : X X is an inclusion of T 0 -spaces, then every continuous map f : X S is the restriction of some continuous map g : X S. Hence, g i = f. X f i X S... (iii) A T 0 -space is injective iff it is a retract of a power of S. g Jan Paseka (MU) 10. 8. 2018 3 / 39
1. Introduction: Complete Boolean algebras and injectivity Statement (i) The injective objects in the category Bool of Boolean algebras are precisely the complete Boolean algebras. (ii) The injective objects in the category DLat of distributive lattices are precisely the complete Boolean algebras. C f i D B... g Jan Paseka (MU) 10. 8. 2018 4 / 39
1. Introduction: Quantales and quantale-like structures Definitions Definition 1 (Mulvey C.J., 1986) A structure (Q,, ) is called a quantale if (Q, ) is a -semilattice, (Q, ) is a semigroup, and multiplication distributes over arbitrary joins in both coordinates, that is, ( ) a M = {a m m M}, ( ) M a = {m a m M}, for any a Q, M Q. A quantale is commutative if the binary operation is commutative. Jan Paseka (MU) 10. 8. 2018 5 / 39
1. Introduction: Quantales and quantale-like structures Definitions Each quantale Q is residuated in the following natural way: s a b s a r b a s l b, a, b, s Q. Jan Paseka (MU) 10. 8. 2018 6 / 39
1. Introduction: Quantales and quantale-like structures Definitions A frame (locale) L is a complete lattice such that for any a L, M L. a ( M) = {a m m M}, Jan Paseka (MU) 10. 8. 2018 7 / 39
1. Introduction: Quantale modules Definitions Definition 2 (Cf. [6, 11]) Given a quantale Q, a left quantale module is a pair (A, ), where A is a -lattice and : Q A A such that: 1 q ( S) = s S (q s) for every q Q, S A; 2 ( T ) a = t T (t a) for every a A, T Q; 3 q 1 (q 2 a) = (q 1 q 2 ) a for every a A, q 1, q 2 Q. Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in 2016. Thus, Q-modules can be seen as a fuzzification of complete lattices. Jan Paseka (MU) 10. 8. 2018 8 / 39
1. Introduction: Quantale modules Definitions Definition 2 (Cf. [6, 11]) Given a quantale Q, a left quantale module is a pair (A, ), where A is a -lattice and : Q A A such that: 1 q ( S) = s S (q s) for every q Q, S A; 2 ( T ) a = t T (t a) for every a A, T Q; 3 q 1 (q 2 a) = (q 1 q 2 ) a for every a A, q 1, q 2 Q. Q-modules are equivalent to so called Q-sup-lattices, where Q is a unital (commutative) quantale. This fact was first pointed out by Stubbe in 2006 and later proved in quantale-like setting by Solovyov in 2016. Thus, Q-modules can be seen as a fuzzification of complete lattices. Jan Paseka (MU) 10. 8. 2018 8 / 39
1. Introduction: Quantale algebras Definitions Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ) such that: 1 (A,, ) is a quantale; 2 q (a b) = (q a) b for every a, b A, q Q. Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification of quantales. Jan Paseka (MU) 10. 8. 2018 9 / 39
1. Introduction: Quantale algebras Definitions Definition 3 (Cf. [11], Solovyov) Given a commutative quantale Q, a quantale algebra is a quantale module (A, ) such that: 1 (A,, ) is a quantale; 2 q (a b) = (q a) b for every a, b A, q Q. Q-algebras are equivalent to so called Q-quantales, where Q is a unital (commutative) quantale. Thus, Q-algebras can be seen as a fuzzification of quantales. Jan Paseka (MU) 10. 8. 2018 9 / 39
1. Introduction: Injective objects in categories Definitions Definition 4 Let C be a category and let M be a class of morphisms in C. We recall that an object S from C is M-injective in C provided that for any morphism h : A B in M and any morphism f : A S in C there exists a morphism g : B S in C such that gh = f. A f h M B... S g Jan Paseka (MU) 10. 8. 2018 10 / 39
1. Introduction: M-injective hull Definitions Definition 5 M-essential morphism A morphism η : A B in M is called M-essential if every morphism ψ : B C in C, for which the composite ψη is in M, is itself in M. M-injective hull An object H C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η : S H. M-injective hulls are unique up to isomorphism. S f η M H... I g M Jan Paseka (MU) 10. 8. 2018 11 / 39
1. Introduction: M-injective hull Definitions Definition 5 M-essential morphism A morphism η : A B in M is called M-essential if every morphism ψ : B C in C, for which the composite ψη is in M, is itself in M. M-injective hull An object H C is called an M-injective hull of an object S if H is M-injective and there exists an M-essential morphism η : S H. M-injective hulls are unique up to isomorphism. S f η M H... I g M Jan Paseka (MU) 10. 8. 2018 11 / 39
1. Introduction: Results on injective hulls of posets 1967, Banaschewski B., Bruns G., Categorical construction of the MacNeille completion, Arch. Math. Theorem 1 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent: 1 P is a complete lattice; 2 P is injective in posets with respect to the class of order embeddings; 3 P is a retract of every extension, i.e., for any order embedding j : P R there is an order-preserving map k : R P such that k j = id P ; 4 P has no essential extensions, i.e., for any order embedding j : P R the poset R is isomorphic to P. Jan Paseka (MU) 10. 8. 2018 12 / 39
1. Introduction: Results on injective hulls of posets Theorem 2 (Banaschewski B., Bruns G.) For a partially ordered set P, the following conditions are equivalent: 1 E is a MacNeille completion of P; 2 E is an injective hull of P; 3 E is an injective extension of P not containing any properly smaller such extension of P; 4 E is an essential extension of P not contained in any properly larger such extension of P. Jan Paseka (MU) 10. 8. 2018 13 / 39
1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a M = ( a m m M ), (1) for all a S, M S. Distributive joins We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a S we have a M = ( a m m M ). Jan Paseka (MU) 10. 8. 2018 14 / 39
1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of meet-semilattices, Canadian Mathematical Bulletin Theorem 3 (Bruns G., Lakser H.) A meet-semilattice S is injective iff it is a frame, i.e., it is complete and satisfies a M = ( a m m M ), (1) for all a S, M S. Distributive joins We say that a subset M of a meet-semilattice S has a distributive join if (i) its supremum exists, and (ii) for all a S we have a M = ( a m m M ). Jan Paseka (MU) 10. 8. 2018 14 / 39
1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin I D (S) = {A S A = A ; M A has a distributive join M A}. Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) I D (S). Jan Paseka (MU) 10. 8. 2018 15 / 39
1. Introduction: Results on injective hulls of semilattices 1970, Bruns G., Lakser H., Injective hulls of semilattices. Bulletin I D (S) = {A S A = A ; M A has a distributive join M A}. Theorem 4 (Bruns G., Lakser H.) The injective hull of a meet-semilattice S is (up to isomorphism) I D (S). Jan Paseka (MU) 10. 8. 2018 15 / 39
1. Introduction: Results on injective hulls of certain S-systems over a semilattice 1972, Johnson C.S., J.R., McMorris F.R., Injective hulls of certain S-systems over a semilattice. Proc. Amer. Math. Soc. Theorem 5 (Johnson C.S., J.R., McMorris F.R.) The injective hull of an S-system M S is (up to isomorphism) I D (M S ). Jan Paseka (MU) 10. 8. 2018 16 / 39
1. Introduction: backgrounds and motivations 1974, Schein B.M. Injectives in certain classes of semigroups. Semigroup Forum. Theorem 6. (1974 Schein) The category of semigroups has only trivial injectives. Jan Paseka (MU) 10. 8. 2018 17 / 39
1. Introduction: Results on injective hulls of posemigroups 2012, Lambek J., Barr M., Kennison J.F. and Raphael R., Injective hulls of partially ordered monoids. Theory Appl. Categ. The category of po-monoids Partially ordered monoids (po-monoid) with submultiplicative order-preserving mappings, i.e., an order-preserving mapping φ : A B of po-monoids satisfying φ(1) = 1, for all a, b A. φ(a) φ(b) φ(a b), Jan Paseka (MU) 10. 8. 2018 18 / 39
1. Introduction: Results on injective hulls of posemigroups 2012, Lambek J., Barr M., Kennison J.F. and Raphael R., Injective hulls of partially ordered monoids. Theory Appl. Categ. Theorem 7 (2012 Lambek, Barr, Kennison and Raphael) A po-monoid (S, ) is injective iff it is a quantale, i.e., it is complete and satisfies a M = ( a m m M ), (2) for all a S, M S.. Jan Paseka (MU) 10. 8. 2018 19 / 39
2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) 10. 8. 2018 20 / 39
2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) 10. 8. 2018 20 / 39
2. Injective constructions for posemigroups (Definitions) Category Pos A posemigroup (S,, ) is both a semigroup (S,, ) and a poset (S, ) such that for any a, b, c, d S, a b, c d = a c b d. Morphisms in posemigroups: order-preserving submultiplicative mappings (subhomomorphisms), i.e., for all a 1, a 2 S. f(a 1 ) f(a 2 ) f(a 1 a 2 ) We denote the category of posemigroups with subhomomorphisms as morphisms by Pos. Jan Paseka (MU) 10. 8. 2018 20 / 39
2. Injective constructions for posemigroups (Definitions) Embeddings in Pos Let ε be the class of morphisms e : S T in the category Pos which satisfy the following condition: for all a 1, a 2,..., a n S. e(a 1 )... e(a n ) e(a) = a 1... a n a, Jan Paseka (MU) 10. 8. 2018 21 / 39
2. Injective constructions for posemigroups (Examples) An example (Rosenthal) Let (S,, ) be a posemigroup, P(S) the set of all of the downsets in S. Then (P(S),, ) is a posemigroup, where X Y = (X Y ) = {a S a x y, for some x X, y Y }, for all X, Y S. Moreover, P(A) is a complete lattice whose joins are unions. Consequently, we obtain that (P(S),, ) is a quantale. Jan Paseka (MU) 10. 8. 2018 22 / 39
2. Injective constructions for posemigroups (Injectives) Theorem 8 (2014 Zhang, Laan) For a posemigroup S, the following statements are equivalent: 1 S is ε -injective in Pos, 2 S is a quantale. Jan Paseka (MU) 10. 8. 2018 23 / 39
2. Injective constructions for posemigroups (Constructions) Definition: Nucleus A closure operator j on a posemigroup S is called a nucleus if it is a subhomomorphism. We denote S j = {a S a = j(a)}. Proposition Let (S,, ) be a posemigroup, j a nucleus on it. Then (S j, j, ) is again a posemigroup equipped with the multiplication and induced order as a j b = j(a b), for any a, b S. In addition, if S is a quantale, then (S j, j, ) is a quantale as well. Jan Paseka (MU) 10. 8. 2018 24 / 39
2. Injective hulls for posemigroups (Constructions) the nucleus cl: Zhang, Laan 2014, Xia, Zhao, Han 2017 Let (S,, ) be a posemigroup, D P(S). Define cl(d) := {x S ( a, c S 1, b S) a D c b = a x c b}, where S 1 is the monoid obtained from the semigroup S by externally adjoining the identity element 1. Then cl : P(S) P(S) is a nucleus on P(S) satisfying cl(x ) = x, x S. Jan Paseka (MU) 10. 8. 2018 25 / 39
2. Injective hulls for posemigroups Theorem Theorem 9 (Zhang, Laan 2014, Xia, Zhao, Han 2017) Let (S,, ) be a posemigroup, cl: P(S) P(S) be defined as above. Then (P(S) cl, cl, ) is the ε -injective hull of the posemigroup S. Jan Paseka (MU) 10. 8. 2018 26 / 39
3. Injective constructions for S-semigroups (Definitions) (S, ) is always a posemigroup A posemigroup (A,, ) together with a mapping S A A (under which a pair (s, a) maps to an element of A denoted by s a) is called an S-semigroup, denoted by S A, or simple A, if for any a, b A, s, t S, 1 s (a b) = (s a) b = a (s b), 2 (s t) a = s (t a), fulfilling a b, s t = s a t b. Jan Paseka (MU) 10. 8. 2018 27 / 39
3. Injective constructions for S-semigroups (Definitions) Morphisms in Ssgr An order-preserving mapping f : S A S B of S-semigroups is called a subhomomorphism if it is both submultiplicative in posemigroups, i.e., f(a 1 ) f(a 2 ) f(a 1 a 2 ) for all a 1, a 2 A, and S-submultiplicative in S-posets, i.e., s f(a) f(s a) for all a A, s S. We denote the category of S-semigroups with subhomomorphisms as morphisms by Ssgr. Jan Paseka (MU) 10. 8. 2018 28 / 39
3. Injective constructions for S-semigroups (Definitions) Embeddings in Ssgr Let ε 0 be the class of morphisms e : SA S B in the category Ssgr which satisfy the following conditions: and s (e(a 1 )... e(a n )) e(a) = s (a 1... a n ) a, for all a 1, a 2,..., a n, a A, s S. e(a 1 )... e(a n ) e(a) = a 1... a n a, Then ε ε 0, where ε is the class of all S-semigroup homomorphisms that are order-embeddings. Jan Paseka (MU) 10. 8. 2018 29 / 39
3. Injective constructions for S-semigroups (Definitions) A new quantale-like structure An S-semigroup quantale is an S-semigroup ( S A,, ) such that 1 (A,, ) is a quantale; 2 s M = {s m m M}, for every s S, M A. Jan Paseka (MU) 10. 8. 2018 30 / 39
3. Injective constructions for S-semigroups An example Let ( S A,, ) be an S-semigroup, P(A) the set of all of the downsets in A. Then ( S P(A),,, ) is an S-semigroup, where X Y = (X Y ) = {a A a x y, for some x X, y Y }, s X = (s X ) = {a A a s x, for some x X }, for all X, Y A, s S. Moreover, S P(A) is a complete lattice whose joins are unions. Consequently, we obtain that ( S P(A),,, ) is an S-semigroup quantale. Jan Paseka (MU) 10. 8. 2018 31 / 39
3. Injective constructions for S-semigroups (Injectives) Theorem 8 (2017 Zhang, Paseka) For an S-semigroup S A, the following statements are equivalent: 1 2 3 SA is ε 0 -injective in Ssgr, SA is ε-injective in Ssgr, S A is an S-semigroup quantale. Jan Paseka (MU) 10. 8. 2018 32 / 39
3. Injective constructions for S-semigroups (Constructions) Definition: Nucleus A closure operator j on an S-semigroup S A is called a nucleus if it is a subhomomorphism. We denote A j = {a A a = j(a)}. Proposition Let ( S A,, ) be an S-semigroup, j a nucleus on it. Then (A j, j, j ) is again an S-semigroup equipped with the multiplication and action as a j b = j(a b), s j a = j(s a), for any a, b A, s S. In addition, if A is an S-semigroup quantale, then (A j, j, j ) is an S-semigroup quantale as well. Jan Paseka (MU) 10. 8. 2018 33 / 39
3. Injective hulls for S-semigroups (Constructions) the nucleus cl: Let ( S A,, ) be an S-semigroup, D S P(A). Define cl(d) := {x A ( a, c A 1, b A, s S) a D c b = a x c b, s D b = s x b}, where A 1 is the monoid obtained from the semigroup A by externally adjoining the identity element 1. Then cl : P(A) P(A) is a nucleus on SP(A) satisfying cl(x ) = x, x A. Jan Paseka (MU) 10. 8. 2018 34 / 39
3. Injective hulls for S-semigroups Theorem Theorem 9 (2017 Zhang, Paseka) Let ( S A,, ) be an S-semigroup, cl: S P(A) S P(A) be defined as above. Then (P(A) cl, cl, cl, ) is the ε 0 -injective hull of the S-semigroup S A. Jan Paseka (MU) 10. 8. 2018 35 / 39
Conclusion Remarks We have described injectives in the category Ssgr of S-semigroups and showed that every S-semigroup has an ε 0 -injective hull. Based on these results the next step in the future would be to obtain corresponding results in the category of residuated S-semigroups. Jan Paseka (MU) 10. 8. 2018 36 / 39
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Thank you! Jan Paseka (MU) 10. 8. 2018 39 / 39