Topic
Cancellative semigroup
About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.
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01 Dec 1983
TL;DR: This work considers the computation of finite semigroups using unbounded fan-in circuits, finding that there are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset.
Abstract: We consider the computation of finite semigroups using unbounded fan-in circuits. There are constant-depth, polynomial size circuits for semigroup product iff the semigroup does not contain a nontrivial group as a subset. In the case that the semigroup in fact does not contain a group, then for any primitive recursive function f, circuits of size O(nf−1(n)) and constant depth exist for the semigroup product of n elements. The depth depends upon the choice of the primitive recursive function f. The circuits not only compute the semigroup product, but every prefix of the semigroup product. A consequence is that the same bounds apply for circuits computing the sum of two n-bit numbers.
96 citations
TL;DR: In this paper, it was shown that if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals, the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.
Abstract: By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.
95 citations
TL;DR: In this paper, it was shown that a semigroup can be embedded in an ilpotent group of class c if, and only if, it is cancellative and satisfies a certain law L c.
Abstract: It is shown that a semigroup can be embedded in an ilpotent group of class c if, and only if, it is cancellative and satisfies a certain law L c .
87 citations
84 citations
TL;DR: For both extremal results, it is shown that a 3-graph with almost as many edges as the extremal example is approximately tripartite, and stability theorems are analogous to the Simonovits stability theorem for graphs.
83 citations