On Semigroups with Lower Semimodular Lattice of Subsemigroups

The question of which semigroups have lower semimodular lattice of subsemigroups has been open since the early 1960s, when the corresponding question was answered for modularity and for upper semimodularity. We provide a characterization of such semigroups in the language of principal factors. Since it is easily seen (and has long been known) that semigroups for which Green\u27s relation J is trivial have this property, a description in such terms is natural. 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 D-trivial

Generalising quasinormal subgroups

In Cossey and Stonehewer ['On the rarity of quasinormal subgroups', Rend. Semin. Mat. Univ. Padova 125 (2011), 81-105] it is shown that for any odd prime p and integer n >= 3, there is a finite p-group G of exponent p(n) containing a quasinormal subgroup H of exponent p(n-1) such that the nontrivial quasinormal subgroups of G lying in H can have exponent only p, p(n-1) or, when n >= 4, p(n-2). Thus large sections of these groups are devoid of quasinormal subgroups. The authors ask in that paper if there is a nontrivial subgroup-theoretic property X: of finite p-groups such that (i) X is invariant under subgroup lattice isomorphisms and (ii) every chain of X-subgroups of a finite p-group can be refined to a composition series of X-subgroups. Failing this, can such a chain always be refined to a series of X-subgroups in which the intervals between adjacent terms are restricted in some significant way? The present work embarks upon this quest

Semidistributive Inverse Semigroups, II

The description by Johnston-Thom and the second author of the inverse semigroups S for which the lattice LJ(S) of full inverse subsemigroups of S is join semidistributive is used to describe those for which (a) the lattice L(S) of all inverse subsemigroups or (b) the lattice lo(S) of convex inverse subsemigroups have that property. In contrast with the methods used by the authors to investigate lower semimodularity, the methods are based on decompositions via GS, the union of the subgroups of the semigroup (which is necessarily cryptic)