Semigroups with conditions on one-sided congruences

Kozhukhov, I. B.

doi:10.1070/rm1998v053n06abeh000099

Cited by 6 publications

(9 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An affirmative answer has been proved under the assumption of ascending chain condition on both right and left congruences [14]. For some further observations see [17, Problem 1].…”

Section: Theoremmentioning

confidence: 97%

See 1 more Smart Citation

Direct limits and fixed point sets

Bergman

2005

Journal of Algebra

View full text Add to dashboard Cite

For which groups G is it true that whenever we form a direct limit of G-sets, dirlim_{i\in I} X_i, the set of its fixed points, (dirlim_I X_i)^G, can be obtained as the direct limit dirlim_I(X_i^G) of the fixed point sets of the given G-sets? An easy argument shows that this holds if and only if G is finitely generated. If we replace ``group G'' by ``monoid M'', the answer is the less familiar condition that the improper left congruence on M be finitely generated. Replacing our group or monoid with a small category E, the concept of set on which G or M acts with that of a functor E --> Set, and the concept of fixed point set with that of the limit of a functor, a criterion of a similar nature is obtained. The case where E is a partially ordered set leads to a condition on partially ordered sets which I have not seen before (pp.23-24, Def. 12 and Lemma 13). If one allows the {\em codomain} category Set to be replaced with other categories, and/or allows direct limits to be replaced with other kinds of colimits, one gets a vast area for further investigation.Comment: 28 pages. Notes on 1 Aug.'05 revision: Introduction added; Cor.s 9 and 10 strengthened and Cor.10 added; section 9 removed and section 8 rewritten; source file re-formatted for Elsevier macros. To appear, J.Al

show abstract

“…An affirmative answer has been proved under the assumption of ascending chain condition on both right and left congruences [14]. For some further observations see [17, Problem 1].…”

Section: Theoremmentioning

confidence: 97%

“…To approach more systematically the problem of characterizing monoids that satisfy (8), let us recall a useful heuristic for generalizing results about groups G and G-sets to monoids M and M-sets: (14) Groups : normal subgroups : subgroups :: monoids : congruences : left congruences.…”

Section: Left Congruences and A Precise Criterionmentioning

confidence: 99%

Direct limits and fixed point sets

Bergman

2005

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…If a monoid M has ascending chain condition on left congruences, must M be finitely generated? It has been shown [103] that if M has ascending chain condition on both left and right congruences, then it is indeed finitely generated. (Students wishing to read that paper might want to have the standard reference work [68] in hand, since semigroup theorists use some rather idiosyncratic terminology.)…”

Section: Definition 918 Ifmentioning

confidence: 99%

An Invitation to General Algebra and Universal Constructions

Bergman

2015

Universitext

View full text Add to dashboard Cite

“…Related to the property of being weakly right noetherian is the stronger condition that every right congruence is finitely generated; we call semigroups satisfying this condition right noetherian . Such semigroups were studied systematically in [21] and had previously been considered in [12, 17, 18]. Another related notion is that of the universal right congruence being finitely generated, which was first considered in [5].…”

Section: Introductionmentioning

confidence: 99%

Semigroups whose right ideals are finitely generated

Miller¹

2021

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

show abstract

Semigroups with conditions on one-sided congruences

Cited by 6 publications

References 0 publications

Direct limits and fixed point sets

Direct limits and fixed point sets

An Invitation to General Algebra and Universal Constructions

Semigroups whose right ideals are finitely generated

Contact Info

Product

Resources

About