N-Congruences of N-semigroups
TL;DR: In this article, a congruence ϱ on an N -semigroup S is called an N-congruence on S if S/g9 is an n-semigroup.
About: This article is published in Journal of Algebra.The article was published on 1973-10-01 and is currently open access. It has received 4 citations till now. The article focuses on the topics: Cancellative semigroup & Semigroup.
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01 Jan 1975
TL;DR: In this paper, it was shown that any maximal cancellative subsemigroup T of a commutative, idempotent-free, archimedean semigroup S must be a mild ideal of S. It is also shown that if T is mild ideal, then every cancellative congruence of T has a unique extension to a cancellative CONGUE of S, and that T is also a semigroup that can be seen as an extension to S.
Abstract: A subsemigroup T of a commutative semigroup S is called a mild ideal if for any a E S, aT n T / 0. It is shown here that any maximal cancellative subsemigroup T of a commutative, idempotentfree, archimedean semigroup S must be a mild ideal of S. Maximal cancellative subsemigroups exist in abundance due to Zorn's lemma. It is also shown that if T is mild ideal of a commutative semigroup S, then every cancellative congruence of T has a unique extension to a cancellative congruence of S. 1. Maximal cancellative subsemigroups. Let S be a commutative archimedean semigroup with no idempotents. Let A be a cancellative subsemigroup of S. By the Hausdorff maximal principle (Zorn's lemma), there will exist a maximal 1 cancellative subsemigroup T such that A C T. In particular if a E 5, then the cyclic semigroup (a) is cancellative, and hence there exists a maximal cancellative subsemigroup of S containing a. In what follows, Z + denotes the set of positive integers. We start with Lemma 1. 1. Let S be a commutative, archimedean, idempotent-free semigroup and let T be a maximal cancellative subsemigroup of S. Then for any a E S\T, there exists i E Z+ and t , t2 E T1, u E T, such that ait u = t2u but a' t 4 t2. Proof. We use, without further comment, a result of Tamura (see [21 or [31) that for any a, b E 5, ab 4 b. Now let a E S\T. By maximality of T, the semigroup generated by a and T is not cancellative. So there exist nonnegative integers j, k and t1, t2 E T1 x E 5, such that a't1 7 a t2; 1 t x= a t2x. If j= k, then t a x = t2a x. Since S is archimedean, Received by the editors November 16, 1973. AMS (MOS) subject classifications (1970). Primary 20M10.
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TL;DR: In this article, the authors give necessary and sufficient conditions in the general case from various points of view: embeddings, partial orderings, the system of ideals, B -condition and structure semigroups.
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References
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Book•
01 Jan 1964
TL;DR: A survey of the structure and representation theory of semi groups is given in this article, along with an extended treatment of the more important recent developments of Semi Group Structure and Representation.
Abstract: This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into the theories of minimal ideals in a semi group, inverse semi groups, simple semi groups, congruences on a semi group, and the embedding of a semi group in a group. Among the more important recent developments of which an extended treatment is presented are B. M. Sain's theory of the representations of an arbitrary semi group by partial one-to-one transformations of a set, L. Redei's theory of finitely generated commutative semi groups, J. M. Howie's theory of amalgamated free products of semi groups, and E. J. Tully's theory of representations of a semi group by transformations of a set. Also a full account is given of Malcev's theory of the congruences on a full transformation semi group.
3,533 citations
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TL;DR: In this article, the authors generalize the results of Tamura and Kimura to medial semigroups and prove that the congruence classes of cnr are Archimedean semigroup.
62 citations
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