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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TL;DR: In this article, it was shown that the inverse semigroups of some 2×2 matrices are groups and that the idempotents of such a semigroup constitute subsemilattice of a finite Boolean lattice.
Abstract: We discuss some fundamental properties of inverse semigroups of matrices,and prove that the idempotents of such a semigroup constitute subsemilattice of a finite Boolean lattice, and that the inverse semigroups of some matrices with the same rank are groups.At last,we determine completely the construction of the inverse semigroups of some 2×2 matrices:such a semigroup is isomorphic to a linear group of dimension 2 or a null-adjoined group,or is a finite semilattice of Abelian linear groups of finite dimension,or satisfies some other properties. The necessary and sufficient conditions are given that the sets consisting of some 2×2 matrices become inverse semigroups.
2 citations
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TL;DR: A ring (R,*) with involution * is called formally complex if implies that all Ai are 0 as discussed by the authors, and a semigroup ring (S, *) with proper involution is a formally complex ring.
Abstract: A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.
2 citations
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TL;DR: In this article, a semigroup model of social networks is presented, where data matrices representing the perceived relationships between members of a social network are used to construct a (possibly infinite) data semigroup of derived relations defined by real matrix multiplication.
Abstract: This paper addresses the development of a semigroup model of social networks. Data matrices which represent the perceived relationships between members of a social network are used to construct a (possibly infinite) data semigroup of derived relations defined by (real) matrix multiplication. This complex structure is analyzed by forming interaction semigroups. These semigroups are homomorphic images of the data semigroup. The corresponding congruences are generated by identifying products of finite order which are highly positively correlated. Several methods of generating the interaction semigroups are examined and are shown to generate nonhomomorphic semigroups. For each congruence, an associated triple of numbers can be defined which may serve as an indicator of the validity and/or a measure of the stability of the semigroup model. A series of hypothetical examples is developed to study how the algebraic properties of interaction semigroups reflect and uncover properties of associated networks. Specifi...
2 citations
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2 citations
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TL;DR: In this paper, it was shown that Kobayashi's condition 121 is necessary and sufficient for Ho to be extended to S. The boundedness condition is introduced to ensure that the extension be unique.
Abstract: Let S be a commutative cancellative semigroup and To be a cofinal subsemigroup of S. Let ho be a homomorphism of To into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi's condition 121 is necessary and sufficient for ho to be extended to S. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the "boundedness condition" is introduced. For further study, several examples are given.
2 citations