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r-notion of conjugacy in partial transformation semigroup
Aftab Hussain Shah,Mohd Rafiq Parray 강원경기수학회 2022 한국수학논문집 Vol.30 No.1
In this paper, we present a new definition of conjugacy that can be applied to an arbitrary semigroup and it does not reduce to the universal relation in semigroups with a zero. We compare the new notion of conjugacy with existing notions, characterize the conjugacy in subsemigroups of partial transformations through digraphs and restrictive partial homomorphisms.
IDENTITIES PRESERVED UNDER EPIS OF PERMUTATIVE POSEMIGROUPS
Shah, Aftab Hussain,Bano, Sakeena,Ahanger, Shabir Ahmad The Kangwon-Kyungki Mathematical Society 2022 한국수학논문집 Vol.30 No.2
In 1985, Khan gave some sufficient conditions on semigroup identities to be preserved under epis of semigroups in conjunction with the general semigroup permutation identity. But determination of all identities which are preserved under epis in conjunction with the general permutation identity is an open problem in the category of all semigroups and hence, in the category of all posemigroups. In this paper, we first find some sufficient conditions on an identity to be preserved under epis of posemigroups in conjunction with any nontrivial general permutation identity. We also find some sufficient conditions on posemigroup identities to be preserved under epis of posemigroups in conjunction with the posemigroup permutation identity, not a general permutation identity.
EPIS, DOMINIONS AND ZIGZAG THEOREM IN COMMUTATIVE GROUPS
Shah, Aftab Hussain,Nabi, Muneer,Ahanger, Shabir Ahmad The Kangwon-Kyungki Mathematical Society 2022 한국수학논문집 Vol.30 No.3
In this paper, we introduce the notion of tensor product in groups and prove its existence and uniqueness. Next, we provide the Isbell's zigzag theorem for dominions in commutative groups. We then show that in the category of commutative groups dominions are trivial. This enables us to deduce a well known result epis are surjective in the category of commutative groups.