15 research outputs found
Lattice isomorphisms of bisimple monogenic orthodox semigroups
Using the classification and description of the structure of bisimple
monogenic orthodox semigroups obtained in \cite{key10}, we prove that every
bisimple orthodox semigroup generated by a pair of mutually inverse elements of
infinite order is strongly determined by the lattice of its subsemigroups in
the class of all semigroups. This theorem substantially extends an earlier
result of \cite{key25} stating that the bicyclic semigroup is strongly lattice
determined.Comment: Semigroup Forum (published online: 15 April 2011
Semigroups with certain finiteness conditions and Chernikov groups
The main purpose of this short survey is to show how groups of special structure, which are accepted to be called Chernikov groups, appeared in the considerations of semigroups with certain finiteness conditions. A structure of groups with several such conditions has been described (they turned out to be special types of Chernikov groups). Lastly, a question concerning some special type of Chernikov groups is recalled; this question was raised by the author more than 35 years ago, and it is still open
Modular and lower-modular elements of lattices of semigroup varieties
The paper contains three main results. First, we show that if a commutative semigroup variety is a modular element of the lattice Com of all commutative semigroup varieties then it is either the variety COM of all commutative semigroups or a nilvariety or the join of a nilvariety with the variety of semilattices. Second, we prove that if a commutative nilvariety is a modular element of Com then it may be given within COM by 0-reduced and substitutive identities only. Third, we completely classify all lower-modular elements of Com. As a corollary, we prove that an element of Com is modular whenever it is lower-modular. All these results are precise analogues of results concerning modular and lower-modular elements of the lattice of all semigroup varieties obtained earlier by Ježek, McKenzie, Vernikov, and the author. As an application of a technique developed in this paper, we provide new proofs of the 'prototypes' of the first and the third our results. © 2012 Springer Science+Business Media, LLC