On injective constructions of S -semigroups Jan Paseka Masaryk - PowerPoint PPT Presentation

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 scnu Jan Paseka (MU) 10. 8. 2018 0 / 39

Contents 1 Introduction M Motivation S Sierpi´ nski 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 scnu 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. scnu Jan Paseka (MU) 10. 8. 2018 2 / 39

1. Introduction: Sierpi´ nski 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 . i ✲ X X ′ ✲ . . . . . . . . . . . . . . . . . . . . . f . . g . . . . . . . . . . ✲ ✛ S scnu (iii) A T 0 -space is injective iff it is a retract of a power of S . 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. i ✲ D C ✲ . . . . . . . . . . . . . . . . . . . . . f . g . . . . . . . . . . ✲ ✛ B scnu 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. scnu 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 . scnu 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 � � a ∧ ( M ) = { a ∧ m | m ∈ M } , for any a ∈ L , M ⊆ L . scnu 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: q ∗ ( � S ) = � s ∈ S ( q ∗ s ) for every q ∈ Q , S ⊆ A ; 1 ( � T ) ∗ a = � t ∈ T ( t ∗ a ) for every a ∈ A , T ⊆ Q ; 2 q 1 ∗ ( q 2 ∗ a ) = ( q 1 ⊗ q 2 ) ∗ a for every a ∈ A , q 1 , q 2 ∈ Q . 3 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. scnu 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: q ∗ ( � S ) = � s ∈ S ( q ∗ s ) for every q ∈ Q , S ⊆ A ; 1 ( � T ) ∗ a = � t ∈ T ( t ∗ a ) for every a ∈ A , T ⊆ Q ; 2 q 1 ∗ ( q 2 ∗ a ) = ( q 1 ⊗ q 2 ) ∗ a for every a ∈ A , q 1 , q 2 ∈ Q . 3 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. scnu 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. scnu 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. scnu 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 . h ∈ M ✲ B A ✲ . . . . . . . . . . . . . . . . . . . . . . f . g . . . . . . . . . . ✛ ✲ S scnu 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. η ∈ M ✲ H S ✲ . ✲ . . . . . . . . . . . . . . . . . M . . . . f . ∈ . . . . . g . . . . . ✛ scnu ✲ I 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. η ∈ M ✲ H S ✲ . ✲ . . . . . . . . . . . . . . . . . M . . . . f . ∈ . . . . . g . . . . . ✛ scnu ✲ I 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: P is a complete lattice ; 1 P is injective in posets with respect to the class of order embeddings ; 2 P is a retract of every extension, i.e., for any order embedding j : P → R 3 there is an order-preserving map k : R → P such that k ◦ j = id P ; P has no essential extensions, i.e., for any order embedding j : P → R the 4 poset R is isomorphic to P . scnu 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: E is a MacNeille completion of P ; 1 E is an injective hull of P ; 2 E is an injective extension of P not containing any properly smaller such 3 extension of P ; E is an essential extension of P not contained in any properly larger such 4 extension of P . scnu Jan Paseka (MU) 10. 8. 2018 13 / 39