Monoids over which products of indecomposable acts are indecomposable

Abstract

In this paper we prove that for a monoid S, products of indecomposable right S-acts are indecomposable if and only if S contains a right zero. Besides, we prove that subacts of indecomposable right S-acts are indecomposable if and only if S is left reversible. Ultimately, we prove that the one element right S-act ΘS is product flat if and only if S contains a left zero.

Monoid Properties as Invariants of Toposes of Monoid Actions

“…51 In the above we recovered Sedaghatjoo and Khaksari's result[37, Lemma 3.7], which is the equivalence (1 ⇔ 3). The equivalence (4 ⇔ 5) can also be seen as the statement that the left M-set with one element is pullback-flat if and only if it is equalizer-flat.…”

“…(6 ⇒ 7 ⇒ 8 ⇒ 9) These implications are trivial. The equivalence of conditions 1 and 7 appears as Proposition 3.9 of [37]; we underline once again that their 'right zero' elements are our 'left absorbing' elements.…”

“…For the global sections morphisms of the toposes studied in this paper, however, it is equally strong: For our investigation of power-connectedness, we begin with the case where M fails to satisfy the right Ore condition. In [37,Proposition 3.4] it is noted that a further condition coinciding with the right Ore condition is the property that every indecomposable M-set has finite width, in the sense that there exists an upper bound on the length of the zigzag needed to connect any pair of elements. This is formalised as follows: Definition 2.43 ([37, Section 1]) Let M be a monoid and A a right M-set.…”

“…31 A monoid M is said to satisfy the right Ore condition or is left reversible if for every m 1 , m 2 ∈ M there exists m 1 , m 2 ∈ M with m 1 m 1 = m 2 m 2 , or equivalently m 1 M ∩m 2 M = ∅. We employ the former terminology, as used by Johnstone in [25, Example A2.1.11]; the latter is employed by Sedaghatjoo and Khaksari in[37] and byKilp et al in[26].…”

“…48 (see[37],[26, III, Definition 14.1]) We say that M is right collapsible if for any pair m 1 , m 2 of elements of M, there exists m ∈ M with m 1 m = m 2 m, that is, if M is cofiltered as a category. Theorem 2.49 (Conditions for PSh(M) to be totally connected) The following are equivalent: 1.…”