On semigroups with lower semimodular lattice of subsemigroups.
The question of which semigroups have a lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. The author provides a characterization of such semigroups in the language of principal factors. Since it is easily seen that semigroups for which Green's relation \(\mathcal J\) is trivial have this property, a description in such terms is natural. A semigroup \(S\) has a lower semimodular lattice of subsemigroups if and only if {\parindent=8mm \begin{itemize} \item[(I)]each non-null principal factor of \(S\) is either: \begin{itemize} \item[(a)]a group with lower semimodular subgroup lattice or a singular band; such a semigroup with zero adjoined; or, up to isomorphism, the five-element combinatorial Brandt semigroup \(B_2\); or \item[(b)]a \(\mathcal D\)-trivial idempotent-free [0-]simple semigroup; \end{itemize} \item[(II)]for any nontrivial subgroup of \(S\), with identity \(e\), say, if \(ea\in H_e\) for some \(a\in S\), then either \(e\in\langle a\rangle\) or \(e=ea\); \item[(III)]for each element \(x\) of \(S\) that does not belong to a nontrivial subgroup of \(S\), \begin{itemize} \item[(a)]if \(x=xab\) for some \(a,b\in S\), then either \(x\in\langle a,b\rangle\) or \(x=xa\); and dually, \item[(b)]if the associated principal factor is null and \(x=bxa\) for some \(a,b\in S\), then either \(x\in\langle a,b\rangle\) or \(x=xa\), \item[(c)]if the associated principal factor is of type I(b) and \(x=a_0xa_1x\cdots xa_n\) for some \(a_0,\dots,a_n\in S^1\) and \(n\geq 1\), then either \(x\in\langle a_0,\dots,a_n\rangle\) or \(n=1\) and \(x=xa_1\). \end{itemize} \end{itemize}} In the case of periodic semigroups -- a case that turns out to include all eventually regular semigroups -- the characterization becomes quite explicit and yields interesting consequences. In the general case, it remains an open question whether there exists a simple, but not completely simple, semigroup with this property. Any such semigroup must at least be idempotent-free and \(\mathcal D\)-trivial. Lower semimodularity of the lattice of subsemigroups is inherited by homomorphic images within the class of periodic semigroups. The variety of semigroups all of whose members have the property that their subsemigroup lattices are lower semimodular is called LSM variety. Every LSM variety of semigroups is periodic. The regular LSM varieties are precisely the regular \(\mathcal J\)-trivial varieties described above. The irregular LSM varieties are (a) the variety of left zero semigroups; (b) the variety of right zero semigroups; (c) the periodic group varieties all of whose members have lower semimodular subgroup lattice; and (d) varieties comprising nilextensions of members of a variety of type (a), (b) or (c). In conclusion the author presents a series of examples, primarily demonstrating the independence of the hypotheses in the main theorem, also demonstrating that the alternative outcomes stated in some of those hypotheses are each necessary.
scientific article; zbMATH DE number 5836825
scientific article