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Showing papers on "Idempotence published in 1971"



Journal ArticleDOI
TL;DR: In this article, the Jacobson radical of a Jordan algebra has been defined as the maximal ideal consisting entirely of quasi-invertible elements, in analogy with the case of associative algebras.

34 citations


Journal ArticleDOI
TL;DR: In this paper, the Baer semigroup coordinatisation theory of lattices is extended to the case of a semilattice and the concept of a Baer assembly is introduced.
Abstract: : We consider, for a given ordered set E with minimum element O, the semigroup Q of O-preserving isotone mappings on E and examine necessary and sufficient conditions under which an element in which point f belongs to set Q is such that the left (resp. right) annihilator of f in Q is a principal left (resp. right) ideal of Q generated by a particular type of idempotent. The results obtained lead us to introduce the concept of a Baer assembly which we use to extend to the case of a semilattice the Baer semigroup coordinatisation theory of lattices. We also derive a coordinatisation of particular types of semilattice. (Author)

27 citations



Journal ArticleDOI
TL;DR: In this article, it was shown that a finite partial idempotent Latin square can be embedded in a finite idemic Latin square, which is then shown to be Hopfian.

20 citations



Journal ArticleDOI
Abstract: W Sierpifiski has shown [1] that given any set A and any function f:An~A, f can be obtained by an appropriate composition of binary functions In this paper we consider the corresponding problem for idempotent functions; f'A"~A is idempotent iff (a, a a)=a for all aeA The motivation for this investigation is a sequence of papers by G Gratzer and J Plonka [2]-[5] characterizing the number p, of n-ary polynomials in an algebra Satisfactory answers are known for all but idempotent algebras (an algebra is idempotent if each of its operations is idempotent) For a finite idempotent algebra it is easy to calculate the maximum asymptotic rate of growth of thepn'S and it is clear this rate is achieved when all idempotent functions are polynomials Can this rate be achieved by an idempotent algebra with only a finite number of operations? Obviously the answer is yes if every idempotent function can be obtained by an appropriate composition of, say, binary idempotent functions Let IAI be the cardinality of A We shall now prove the following theorem: If IAI>2 then every idempotent function can be obtained by composition of binary idempotent functions; if IAI =2 then every idempotent function can be obtained by composition of ternary idempotent functions but not by binary idempotent functions LEMMA 1: If IAI ~>No then for all n>~O every n-ary idempotent function can be obtained by composition of ternary idempotent functions

11 citations



Journal ArticleDOI
TL;DR: In this paper, a necessary and sufficient condition for an idempotent semigroup with identity element to be linearly orderable is given, where the identity element is an identity element.
Abstract: The purpose of this paper is to give a necessary and sufficient condition for an idempotent semigroup with identity element to be linearly orderable

7 citations


Journal ArticleDOI
TL;DR: In this paper, the number and cardinality of the corresponding equivalence classes of Green's equivalence relations are determined, and the number of idempotent and generalized ǫ elements in ℱn is also determined.
Abstract: Letℱn be the set of all partial functions on ann-element setXn, i.e., the set of all functions whose domain and range are subsets ofXn. Green's equivalence relationsℛ, ℒ, ℋ andℋ are considered, and the number and cardinality of the corresponding equivalence classes are determined. The number of idempotent and generalized idempotent elements inℱn is also determined.

7 citations


Journal ArticleDOI
TL;DR: It is known that for a given integer n there are at most n − 2 pairwise orthogonal idempotent latin squares as discussed by the authors, and for n a prime power there always exists n −2 such squares.
Abstract: It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.

Journal ArticleDOI
TL;DR: In this paper, the identity of compact monothetic semitopological (separately continuous) semigroups is investigated by the methods of harmonic analysis and the pathology is shown to be arbitrarily bad in a sense made precise.
Abstract: Idempotent structure of compact monothetic semitopological (separately continuous) semigroups is investigated by the methods of harmonic analysis. The pathology is shown to be arbitrarily bad in a sense made precise.