Document Type

Article

Language

eng

Publication Date

10-2012

Publisher

Springer

Source Publication

Semigroup Forum

Source ISSN

0037-1912

Original Item ID

doi: 10.1007/s00233-012-9439-6

Abstract

The left restriction semigroups have arisen in a number of contexts, one being as the abstract characterization of semigroups of partial maps, another as the ‘weakly left E-ample’ semigroups of the ‘York school’, and, more recently as a variety of unary semigroups defined by a set of simple identities. We initiate a study of the lattice of varieties of such semigroups and, in parallel, of their two-sided versions, the restriction semigroups. Although at the very bottom of the respective lattices the behaviour is akin to that of varieties of inverse semigroups, more interesting features are soon found in the minimal varieties that do not consist of semilattices of monoids, associated with certain ‘forbidden’ semigroups. There are two such in the one-sided case, three in the two-sided case. Also of interest in the one-sided case are the varieties consisting of unions of monoids, far indeed from any analogue for inverse semigroups. In a sequel, the author will show, in the two-sided case, that some rather surprising behavior is observed at the next ‘level’ of the lattice of varieties.

Comments

Accepted version. Semigroup Forum, Vol. 86, No. 2 (April 2013; 337-361) . DOI. © Springer 2013. Used with permission.

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