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Showing papers on "Idempotence published in 1972"


Journal ArticleDOI
TL;DR: A finitely based equational class of idempotent algebras of type,m, n ≥ 2, is two-based as mentioned in this paper, and any class of ǫ-based equational classes with m ≥ 2 and k ≥ 2 is k-based.
Abstract: A finitely based equational class of idempotent algebras of type ,m, n≥2, is two-based. More generally, any finitely based equational class of idempotent algebras of type withm i≥2 andk≥2 isk-based.

4 citations



Journal ArticleDOI
01 Jan 1972
TL;DR: In this article, the author's example of a groupoid which has only two idempotents (a zero for least element, and an identity for greatest element), a compact neighborhood of the greatest element consisting of power associative elements, and which is not isomorphic to either the real thread or the nil thread is described.
Abstract: Compact, connected, totally ordered, (Hausdorff) topological groupoids, with restrictions on their sets of idempotents and with varying degrees of power associativity assumed, are ex- amined. The paper evolves from the author's example of such a groupoid which has only two idempotents (a zero for least element, and an identity for greatest element), a compact neighborhood of the greatest element consisting of power associative elements, and which is not isomorphic to either the real thread or the nil thread. Another example given has a zero for least element, an idempotent for greatest element, and no other idempotents, and has a compact neighborhood of the greatest element consisting of an associative subgroupoid in which all products are equal to the great- est element. Theorems are given which show that these examples, and one other, in some sense, exhaust the possibilities. Ordered power associated groupoids have been considered by—among many others—J. Aczel (1), L. Fuchs (2), K. H. Hofmann (4), P. S. Mostert (7), and R. J. Warne (10). We shall provide an example which complements some work of Mostert, and a result which improves some- what a result given by Warne. Thanks are due to Professors Hofmann and Mostert who read the original draft and made detailed suggestions for improvement, and also to the referee who suggested major improve- ments. Due to their help we are able to give a more complete account than originally envisaged. The direction of Professor A. D. Wallace during this research is grate- fully acknowledged. A groupoid is a nonvoid Hausdorff space together with a continuous binary operation (multiplication denoted by juxtaposition). An ordered groupoid is the data of an ordered set together with the order topology, in which the multiplication is continuous. An element of a groupoid is power associative if it is contained in an associative subgroupoid. A sect is an ordered groupoid which is compact and connected (of course in the order topology), which has a zero for least element (Ox = xO = 0 whatever x), and an idempotent, e, for the greatest element (with 0 < e).

2 citations


Journal ArticleDOI
01 Feb 1972
TL;DR: In this article, it is shown that any proof of the independence of a given axiom must involve an example with certain restrictions on the cardinality of the cardinal number of the axiom.
Abstract: The independence of the axioms for spans and the independence of the axioms for closure structures are usually taken for granted. In this paper, the author establishes the independence of monotonicity, extensiveness, idempotence, the exchange property, the property of having 0 as a fixed set and two covering properties (cc-character, with ac being some cardinal number, and a covering property with respect to generators). The independence of the axioms for closure structures and spans follow immediately. It is shown that any proof of the independence of a given axiom must involve an example with certain restrictions on the cardinal

2 citations


Journal ArticleDOI
TL;DR: The objective of this communication is to show that Tainiter's technique of characterizing and counting idempotent elements in a symmetric semigroup can be strengthened and generalized to apply to the characterization and counting of all elements of any symmetry semigroup and in fact of any symmetric m-semigroup.

1 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the symmetric algebras (6, 6 ) and (8, 8 ) of broken dilatation symmetry and showed that for each case, only the SU(3) scalar (denoted by w0) is the only idempotent which does not contain the (27) representation of SU (3).

1 citations