Homological finiteness properties of monoids, their ideals and maximal subgroups

Abstract

a b s t r a c tWe consider the general question of how the homological finiteness property left-FP n (resp. right-FP n ) holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. This is done by giving methods for constructing free resolutions of substructures from free resolutions of their containing monoids, and vice versa. In particular, we show that left-FP n is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups, we prove the converse, and as a corollary, show that a completely simple semigroup is of type left-and right-FP n if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type FP n . Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left-FP n if and only if the ideal is of type left-FP n . Applying this result, we obtain necessary and sufficient conditions for a Clifford monoid (and more generally a strong semilattice of monoids) to be of type left-FP n . Examples are provided showing that for each of the results all of the hypotheses are necessary.

Finite complete rewriting systems for regular semigroups

“…In [11,Theorem 6.2] it is proved that A, B, C | R, (4), (5) (where (4) and (5) are defined below) is a presentation for S as a semigroup with zero. This may be converted into a genuine semigroup presentation for S by adding a new generating symbol 0 and relations (8). Adding to this two additional families of redundant relations (6) and (7), we obtain the following presentation for S:…”

“…The easiest case is when a rewriting rule of type (5), (6) or (7) is applied because situation (i) occurs. We distinguish three cases when a rewriting rule of type (8) is applied: if x ∈ B ∪ C we are in situation (i); if x ≡ 0 then situation (ii) occurs; otherwise, if x ∈ A we have (iii).…”

“…In [5] we positively answered this question for the property FDT. The questions regarding finite cohomological dimension, FP n and FP ∞ are discussed in [8] where is it observed that the corresponding result does not hold for any of these properties.…”

Semigroups with finitely generated universal left congruence

“…From a homological standpoint the concept of being of type left-FP n was introduced for groups by Bieri [2] and later extended to monoids [4]. Much of the work into the general property of being of type left-FP n has been in the case of groups, although a recent shift to monoids can be seen in [9,10,15,16,21]. Any monoid which possesses a finite complete rewriting system was shown by Anick [1] to be of type left-FP n for each n. We refer the reader to [5] for a wider study.…”

“…Our results often divide into two kinds: those for semigroups and those for monoids. On the way we consider numerous constructions, some of which have been considered before in the monoid case-see [10] for direct products, [18,22] for retracts and [9] for Clifford monoids. An important word of warning: the property of being of type left-FP 1 does not apply to semigroups.…”