Orthodox semigroups whose idempotents satisfy a certain identity

Abstract

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx = xy [xyx = yx].Bisimple left [right] inverse semigroups have been studied by Venkatesan [6].In this paper, we clarify the structure of general left [right] inverse semigroups.Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx = xyzx.In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx = xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup. I. PRELIMINARIESA

LeftRight Clifford Semigroups

“…By [12], η 1 is a congruence on S and S/η 1 is a left inverse semigroup. Dually, a relation η 2 on S is defined by aη 2 b if and only if…”

“…By [12], S is isomorphic to a subdirect product of S/η 1 and S/η 2 where the isomorphism φ : S −→ S/η 1 × S/η 2 is given by aφ = (ã,ã), a ∈ S whereã,ã denote the η 1 -class and η 2 -class containing a, respectively. Let ψ : S/η 1 −→ S/σ and ξ : S/η 2 −→ S/σ be defined byãψ =ā andãξ =ā, whereā is the σ-class containing a.…”

“…According to Yamada [12], a regular semigroup is a quasi-inverse semigroup if and only if it is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.…”

Note on a certain class of orthodox semigroups

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This is a continuation and also a supplement of the previous papers [5], [6] and [8] concerning orthodox semigroupsllIn [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.In this paper, we present a structure theorem for quasi-inverse semigroups and some relevant matters. e e E, define Seas follows: fse= efe, fs E. Then, (fh)8 e = efhe = efehe = (efe) (ehe) = (f6e) (h~ e) for f,h eE (since E is a regular band).

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“…In [8], it has been shown that a quasi-inverse semigroup is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.…”