Bisimple left inverse semigroups

Venkatesan, P. S.

doi:10.1007/bf02570767

“…A regular semigroup S is called right inverse if for any idempotents e,foiS, efe = fe. Such semigroups have been studied by Ewing (1971), Venkatesan (1972), Bailes (1972) and Warne (1980). Right normal right inverse semigroups form a special class of right inverse semigroups and a faithful representation for these has been given by the author (1976).…”

Section: Introductionmentioning

confidence: 99%

Right inverse semigroups

Madhavan

¹

1980

J Aust Math Soc A

0

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In a recent paper of the author the well-known Vagner-Preston Theorem on inverse semigroups was generalized to include a wider class of semigroups, namely right normal right inverse semigroups. In an attempt to generalize the theorem to include all right inverse semigroups, the notion of μ – μi transformations is introduced in the present paper. It is possible to construct a right inverse band BM(X) of μ – μi transformations. From this a set AM(X) for which left and right units are in BM(X) and satisfying certain conditions is constructed. The semigroup AM(X) so constructed is a right inverse semigroup. Conversely every right inverse semigroup can be isomorphically embedded in a right inverse semigroup constructed in this way.1980 Mathematics subject classification (Amer. Math. Soc.): 20 M 20.

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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. PRELIMINARIES

A

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confidence: 95%

Orthodox semigroups whose idempotents satisfy a certain identity

Yamada¹

1973

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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

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“…semigroup (see [3], [8]). Hereafter, " a band B~{By : 7 e } " means " a band B having the structure decomposition B~Z{ BT: 7 e ~}(see [5])" Any other terminology and notation should be referred to [5], [6] and [8], unless otherwise stated.…”

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confidence: 98%

Note on a certain class of orthodox semigroups

Yamada

¹

1973

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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). Hence, 6 e is an endomorphism on E. This endomorphism 8 e is called the inner endomorphism on E induced by e. Now, let F be an inverse semigroup and A its basic semilattice (see [5]). Let B~ ~{B~ :~e A} be a regular band.i) In [6] and [7], the author used the terminology " a strictly regular semigroup " for " an orthodox semigroup " In this paper, we use the terminology " an orthodox semigroup " introduced by Hall [I].

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Bisimple left inverse semigroups

Cited by 10 publications

References 4 publications

Right inverse semigroups

Right inverse semigroups

Orthodox semigroups whose idempotents satisfy a certain identity

Note on a certain class of orthodox semigroups

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