The Congruence Lattice of a Regular Semigroup

Document Type

Article

Language

eng

Publication Date

8-1988

Publisher

Elsevier

Source Publication

Journal of Pure and Applied Algebra

Source ISSN

0022-4049

Abstract

Let S be a regular semigroup and ConS the congruence lattice of S. If C" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 14.4px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">C is an isomorphism class of semigroups and ϱϵConS, then we say that ϱ is over C" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 14.4px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">C if the idempotent ϱ-class belong to C" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 14.4px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">C. On ConS we can introduce the relations U, V, Tl, Tr and T as follows: if ϱ, θ, ϵConS, then we say that ϱ and θ are U− [V−, Tl−, Tr−, T−] related if both ϱ/ϱ∩θ and θ/ϱ∩gqover completely simple semigroups [rectangular band, left groups, right groups, groups]. It is shown that U, V, Tl, Tr and T are complete congruences on ConS.Various other characterizations of these congruences on ConS are obtained. Some of the congruences are studied for completely regular semigroups, orthodox semigroups and bands of groups. Further, since for any regular semigroup S, V∩Tl∩Tr is the identity relation, we obtain a subdirect decomposition of ConS.

Comments

Journal of Pure and Applied Algebra, Vol. 53, Nos. 1-2 (August 1988): 93-123. DOI.

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