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A Munn type representation of abundant semigroups with a multiplicative ample transversal

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Abstract

The celebrated construction by Munn of a fundamental inverse semigroup \(T_E\) from a semilattice E provides an important tool in the study of inverse semigroups and ample semigroups. Munn’s semigroup \(T_E\) has the property that a semigroup is a fundamental inverse semigroup (resp. a fundamental ample semigroup) with a semilattice of idempotents isomorphic to E if and only if it is embeddable as a full inverse subsemigroup (resp. a full subsemigroup) into \(T_E\). The aim of this paper is to extend Munn’s approach to a class of abundant semigroups, namely abundant semigroups with a multiplicative ample transversal. We present here a semigroup \(T_{(I,\Lambda , E^{\circ }, P)}\) from a so-called admissible quadruple \((I,\Lambda , E^{\circ }, P)\) that plays for abundant semigroups with a multiplicative ample transversal the role that \(T_E\) plays for inverse semigroups and ample semigroups. More precisely, we show that a semigroup is a fundamental abundant semigroup (resp. fundamental regular semigroup) having a multiplicative ample transversal (resp. multiplicative inverse transversal) whose admissible quadruple is isomorphic to \((I,\Lambda , E^{\circ }, P)\) if and only if it is embeddable as a full subsemigroup (resp. full regular subsemigroup) into \(T_{(I,\Lambda , E^{\circ }, P)}\). Our results generalize and enrich some classical results of Munn on inverse semigroups and of Fountain on ample semigroups.

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Acknowledgments

The author expresses his profound gratitude to the referee for the valuable comments, which improve greatly the presentation of this article. In particular, the author shortens the original proof of Theorem 4.4 according to the referee’s suggestions. As the referee has pointed out, ample semigroups are generalized to restriction semigroups now, and a generalized Munn representation for restriction semigroups is also explored in the literature (see [11, 15, 18]). It is likely that the results of this paper could be extended by similar methods to some classes of E-semiabundant semigroups. Thanks also go to Professor Maria B. Szendrei for the timely communications. This paper is supported jointly by a Nature Science Foundation of Yunnan Province (2012FB139) and a Nature Science Foundation of China (11301470).

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  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, Yunnan, People’s Republic of China

    Shoufeng Wang

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  1. Shoufeng Wang
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Wang, S. A Munn type representation of abundant semigroups with a multiplicative ample transversal. Period Math Hung 73, 43–61 (2016). https://doi.org/10.1007/s10998-016-0124-5

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