Abstract
An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR1)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR1), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR1), etc.
References
Clifford, A.H. and Preston, G.B: The Algebraic Theory of Semigroups, Amer. Math. Soc. Surveys, Vol. I, Providence, (1961)
Srinivasan, B.R: Weakly Inverse Semigroups, Math. Annalen, Vol. 176 (1968)
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Communicated by A.H. Clifford
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Namboodripad, K.S.S. On some classes of regular semigroups. Semigroup Forum 2, 264–270 (1971). https://doi.org/10.1007/BF02572293
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DOI: https://doi.org/10.1007/BF02572293