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Quasi Injectivity of Partially Ordered Acts

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Quasi Injectivity of Partially Ordered Acts Quasi injectivity of partially ordered acts Mahdieh Yavari Shahid Quasi injectivity of partially ordered acts Beheshti University Introduction Mahdieh Yavari Preliminaries Shahid Beheshti University E-quasi injectivity in Pos-S E-quasi injectivity and TACL 26-30 June 2017 θ-extensions Charles University Reversiblity and E-quasi injectivity of S-posets References 1/50 ) Partially Ordered Sets with actions of a pomonoid S on them (Pos-S) Introduction Quasi injectivity of partially ordered acts Mahdieh Actions of a Monoid on Sets Yavari Shahid Beheshti University + Introduction Actions of a Monoid on Partially Ordered Sets Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 2/50 Introduction Quasi injectivity of partially ordered acts Mahdieh Actions of a Monoid on Sets Yavari Shahid Beheshti University + Introduction Actions of a Monoid on Partially Ordered Sets Preliminaries E-quasi injectivity in ) Pos-S E-quasi injectivity and Partially Ordered Sets with actions of a pomonoid S θ-extensions on them (Pos-S) Reversiblity and E-quasi injectivity of S-posets References 2/50 Banaschewski: In the category of posets with order preserving maps (Pos) E-injective = Complete posets Introduction Quasi injectivity of partially ordered acts Mahdieh Injectivity Yavari Shahid Injectivity, which is about extending morphisms to a bigger Beheshti University domain of definition, plays a fundamental role in many branches of mathematics. Introduction Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 3/50 Introduction Quasi injectivity of partially ordered acts Mahdieh Injectivity Yavari Shahid Injectivity, which is about extending morphisms to a bigger Beheshti University domain of definition, plays a fundamental role in many branches of mathematics. Introduction Preliminaries E-quasi injectivity in Pos-S Banaschewski: E-quasi In the category of posets with order preserving maps (Pos) injectivity and θ-extensions E-injective = Complete posets Reversiblity and E-quasi injectivity of S-posets References 3/50 Ebrahimi, Mahmoudi, Rasouli: In Pos-S Mono-injective object = Trivial object Pos-S has enough E-injective objects. Introduction Quasi injectivity of partially ordered acts Mahdieh Sikorski: Yavari Shahid Beheshti In the category of BoolAlg University E-injective = Complete Boolean algebras Introduction Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 4/50 Introduction Quasi injectivity of partially ordered acts Mahdieh Sikorski: Yavari Shahid Beheshti In the category of BoolAlg University E-injective = Complete Boolean algebras Introduction Preliminaries E-quasi injectivity in Pos-S Ebrahimi, Mahmoudi, Rasouli: E-quasi In Pos-S Mono-injective object = Trivial object injectivity and θ-extensions Pos-S has enough E-injective objects. Reversiblity and E-quasi injectivity of S-posets References 4/50 Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Ahsan, gave a characterization of monoids whose class of quasi injective S-acts is closed under the formation of direct sums. Also, he investigated monoids all of whose S-acts are quasi injective. Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity. Quasi Injectivity Quasi injectivity of partially ordered acts The study of different kinds of weakly injectivity is an Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University Introduction Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 5/50 Ahsan, gave a characterization of monoids whose class of quasi injective S-acts is closed under the formation of direct sums. Also, he investigated monoids all of whose S-acts are quasi injective. Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity. Quasi Injectivity Quasi injectivity of partially ordered acts The study of different kinds of weakly injectivity is an Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Introduction Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 5/50 Roueentan and Ershad, introduced duo S-act, inspired from the concept of duo module in module theory, which is tightly related to quasi injectivity. Quasi Injectivity Quasi injectivity of partially ordered acts The study of different kinds of weakly injectivity is an Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Introduction Preliminaries Ahsan, gave a characterization of monoids whose class of E-quasi quasi injective S-acts is closed under the formation of injectivity in Pos-S direct sums. Also, he investigated monoids all of whose E-quasi S-acts are quasi injective. injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 5/50 Quasi Injectivity Quasi injectivity of partially ordered acts The study of different kinds of weakly injectivity is an Mahdieh interesting subject for mathematicians. One of the Yavari Shahid important kinds of weakly injectivity is quasi injectivity. Beheshti University Quasi injective S-acts have been studied by Berthiaume, Satyanarayana, Lopez and Luedeman. Introduction Preliminaries Ahsan, gave a characterization of monoids whose class of E-quasi quasi injective S-acts is closed under the formation of injectivity in Pos-S direct sums. Also, he investigated monoids all of whose E-quasi S-acts are quasi injective. injectivity and θ-extensions Roueentan and Ershad, introduced duo S-act, inspired Reversiblity and E-quasi from the concept of duo module in module theory, which injectivity of S-posets is tightly related to quasi injectivity. References 5/50 Preliminaries Quasi injectivity of partially ordered acts Mahdieh Definition (Act-S) Yavari Shahid Beheshti Recall that a right S-act or S-set for a monoid S is a set A University equipped with an action A × S ! A,(a; s) as, such that Introduction (1) ae = a (e is the identity element of S), Preliminaries (2) a(st) = (as)t, for all a 2 A and s; t 2 S. E-quasi injectivity in Pos-S Let Act-S denote the category of all S-acts with action E-quasi preserving maps (f : A ! B with f (as) = f (a)s, for all injectivity and θ-extensions a 2 A; s 2 S). Reversiblity and E-quasi injectivity of S-posets References 6/50 Preliminaries Quasi injectivity of partially ordered acts Mahdieh Yavari Shahid Beheshti University Definition (Zero element) Introduction Let A be an S-act. An element a 2 A is called a zero, fixed, or Preliminaries a trap element if as = a, for all s 2 S. E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 7/50 Definition (Right (Left) Zero Semigroup) A semigroup S is called right (left) zero if its multiplication is defined by st = t(st = s), for all s; t 2 S. Also, S[f_ eg is called a right (left) zero semigroup with an adjoined identity, if S is a right (left) zero semigroup and se = s = es, for all s 2 S, and ee = e. Preliminaries Quasi injectivity of partially ordered acts Definition (Po-monoid, Po-group) Mahdieh A po-monoid (po-group) S is a monoid (group) with a partial Yavari Shahid order ≤ which is compatible with its binary operation (that is, Beheshti 0 0 0 0 0 0 University for s; t; s ; t 2 S, s ≤ t and s ≤ t imply ss ≤ tt ). Introduction Preliminaries E-quasi injectivity in Pos-S E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 8/50 Preliminaries Quasi injectivity of partially ordered acts Definition (Po-monoid, Po-group) Mahdieh A po-monoid (po-group) S is a monoid (group) with a partial Yavari Shahid order ≤ which is compatible with its binary operation (that is, Beheshti 0 0 0 0 0 0 University for s; t; s ; t 2 S, s ≤ t and s ≤ t imply ss ≤ tt ). Introduction Preliminaries E-quasi Definition (Right (Left) Zero Semigroup) injectivity in Pos-S A semigroup S is called right (left) zero if its multiplication is E-quasi _ injectivity and defined by st = t(st = s), for all s; t 2 S. Also, S[feg is θ-extensions called a right (left) zero semigroup with an adjoined identity, if Reversiblity and E-quasi S is a right (left) zero semigroup and se = s = es, for all injectivity of S-posets s 2 S, and ee = e. References 8/50 Preliminaries Quasi injectivity of partially ordered acts Mahdieh Yavari Shahid Beheshti Definition (Pos-S) University For a po-monoid S, a right S-poset is a poset A which is also Introduction an S-act whose action λ : A × S ! A is order-preserving, Preliminaries where A × S is considered as a poset with componentwise E-quasi injectivity in order. The category of all S-posets with action preserving Pos-S monotone maps between them is denoted by Pos-S. E-quasi injectivity and θ-extensions Reversiblity and E-quasi injectivity of S-posets References 9/50 Preliminaries Quasi injectivity of partially ordered acts Mahdieh Yavari Shahid Beheshti Definition (Order-embedding) University A morphism f : A ! B in the category Pos-S and its Introduction subcategories, is called order-embedding, or briefly embedding, Preliminaries if for all x; y 2 A, f (x) ≤ f (y) if and only if x ≤ y.
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