2013
DOI: 10.1017/s0017089513000086
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Ideals and Finiteness Conditions for Subsemigroups

Abstract: In this paper we consider a number of finiteness conditions for semigroups related to their ideal structure, and ask whether such conditions are preserved by sub-or supersemigroups with finite Rees or Green index. Specific properties under consideration include stability, D = J and minimal conditions on ideals.2010 Mathematics Subject Classification. 20M05, 20M12.

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Cited by 3 publications
(3 citation statements)
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References 25 publications
(45 reference statements)
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“…The related question of the existence of finite complete rewriting system has been settled only very recently, see [16]. Some related cohomological finiteness conditions are considered in [11] and [15], residual finiteness is treated in [13], and some surprising behaviour in relation to the ideal structure is exhibited in [3].…”
Section: Introduction and The Statement Of Main Resultsmentioning
confidence: 99%
“…The related question of the existence of finite complete rewriting system has been settled only very recently, see [16]. Some related cohomological finiteness conditions are considered in [11] and [15], residual finiteness is treated in [13], and some surprising behaviour in relation to the ideal structure is exhibited in [3].…”
Section: Introduction and The Statement Of Main Resultsmentioning
confidence: 99%
“…Moreover, subsemigroups of finite Green index preserve finiteness and finite generation (the question of finite presentability is still open). For more details about Green index see [3,8,9]. The analog of Theorem 4.1 for Green index does not hold for arbitrary semigroups, as can be seen from the following example: Proof That T has Green index 2 in S is obvious.…”
Section: Subsemigroups Of Finite Indexmentioning
confidence: 99%
“…In this subsection, we show that the Green index serves as a better analogy of the group index [20] and gives us equality, directly generalizing the result for groups. The Green index, which was introduced in [20], has proven to be a very useful generalization of both the grouptheoretic notion of index and the more established Rees index for semigroups, and has yielded many Reidemeister-Schreier-type theorems about the inheritance of various finiteness properties by subsemigroups or extensions of finite index; see, for example, [7,10,19,20,24,28]. We recall the definition here: let T be a subsemigroup of a semigroup S .…”
Section: 4mentioning
confidence: 99%