| In this thesis, we mainly studyπ-regular semigroups . It can be divided into four chapters.Chapter 1 is a brief introduction.In Chapter 2 , we studyπ-regular semigroups whose idempotents satisfy permutation identities . It is well-known that aπ-regular semigroup in which the set of idempotents satisfies commutativity is a stronglyπ-inverse semigroup. By a generalized stronglyπ-inverse semigroup is meant aπ-regular semigroup in which the set of idempotents satisfies a permutation identity x1x2…xn = xp1xp2…xpn (where (p1,P2,…,pn) is a nontrivial permutition of (1,2,…, n)), its structure theorem and the concept of quasidirect product are given. It contains four sections. Section 1 is a brief introduction. In Section 2, a concept of a generalized stronglyπ-inverse semigroup is introduced. After that, its relative characterizations are given. In Section 3, we study Nπ-inversive semigroups. Section 4 is devoted to classification of permutation identities.Chapter 3 is devoted toπ-regular subsets of semigroups related to their idempotents. For a subsemigroup T of a semigroup S, letπ—Reg(T) andπ—reg(T) denote respectively the set of allπ-regular elements of T and the set of all elements of T which areπ-regular in S. We characterize semigroups withπ—Reg (T) =π—reg(T), where T runs over one of the folowing families of subsemigroups: {Se|e∈E(S)}, {eS|e∈E{S)}, {eSf|e, f∈E(S)}. It contains two sections. Section 1 is a brief introduction. In Chapter 2, we gain main results.In Chapter 4, we study variants ofπ-regular semigroups. Let S be aπ-regular semigroup, and a∈S. Then a variant of S with respect to a is a semigroup with underlying set S and multiplicationοdefined by xοy = xamy, m∈Z such that am is regular. In this Chapter, we characterise theπ-regularity preserving elements ofπ-regular semigroups. It contains four sections. Section 1, we introduce some basic preliminaries about variants ofπ-regular semigroups and some necessary preparations. In Section 2, we give a concept ofπ-regularity preserving elements. In Section 3, we study the structure of variants. Section 4 is devoted to generalized variants ofπ-regular semigroups. That is, let S be aπ-regular semigroup, and a∈S. Then a variant of S with respect to a is a semigroup with underlying set S and multiplicationοdefined by xοy = xay.
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