Showing papers on "Idempotence published in 1978"
TL;DR: In this article, the authors investigated the relationship between group algebra and semigroup amenability for inverse semigroups S and obtained partial results for S with infinite sets of idempotent elements.
Abstract: If G is a group, then G is amenable as a semigroup if and only if l1(G), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S. A complete answer can be given in the case where the set Es of idempotent elements of S is finite. Some partial results are obtained for inverse semigroups S with infinite Es.
137 citations
TL;DR: In this paper, it was shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n-1) members, and a formula was given for the number of distinct sets M. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n 1) idempots whose range has cardinal n − 1 form a generating set for.
Abstract: It was proved by Howie in 1966 that , the semigroup of all singular mappings of a finite set X into itself, is generated by its idempotents. Implicit in the method of proof, though not formally stated, is the result that if |X| = n then the n(n – 1) idempotents whose range has cardinal n – 1 form a generating set for. Here it is shown that if n ≧ 3 then a minimal set M of idempotent generators for contains ½n(n–1) members. A formula is given for the number of distinct sets M.
79 citations
TL;DR: For every field F and every pair (n,k) of positive integers, an n×n matrix S over F is a product of k idempotent matrices over F iff rank(I − S)⩽k· nullity S.
77 citations
TL;DR: In this article, Mendelsohn et al. showed that the existence of semi-symmetric idempotent quasigroups satisfying Steiner triple systems is known.
Abstract: A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schroder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.
19 citations
01 Jan 1978
TL;DR: In this article, the structure of certain semiprimitive rings with involution * is determined by imposing conditions on the set of *-symmetric elements and limiting the number of orthogonal *-Symmetric idempotents.
Abstract: The structure of certain semiprimitive rings with involution * is determined by imposing conditions on the set of *-symmetric elements and limiting the number of orthogonal *-symmetric idempotents. An associative ring R with unit such that 1/2 E R satisfies condition C (n) if (i) R has an involution * such that each *-symmetric element s of R is nilpotent or some (right) multiple of s is a nonzero *-symmetric idempotent and (ii) R has a set of n nonzero, pairwise orthogonal, *-symmetric idempotents whose sum is one and if { e1)7=I is any set of such idempotents whose sum is one, then m < n. We will determine the structure of all rings with condition C (n). A ring satisfying C(l) has exactly one nonzero *-symmetric idempotent, the one of that ring. Hence each *-symmetric element is either nilpotent or invertible. Osborn [4] catalogued these rings and showed that a semiprimitive ring has C(1) iff R is one of (i) a division ring, (ii) a direct sum of two anti-isomorphic division rings with involution interchanging the summands, (iii) the 2 x 2 matrices over a field with the involution fixing only the scalar matrices. First we reduce to the case where R is semiprimitive. Then we collect some of the facts to be used for our main result (Theorem 4). Since the results of Lemma 1 and Lemma 2 are well known, their proofs are omitted. LEMMA 1. If each *-symmetric element s of R is either nilpotent or some (right) multiple of s is a nonzero *-symmetric idempotent, then the Jacobson radical of R, J(R), is a *-invariant ideal in which every *-symmetric element is nilpotent. LEMMA 2. If each *-symmetric element s of R is either nilpotent or some (right) multiple of s is a nonzero *-symmetric idempotent, and uE R/J(R) is a symmetric element under the induced involution *', then uis either nilpotent or some (right) multiple of uis a nonzero *'-symmetric idempotent. REMARK 1. Suppose R is a ring and for some a E R, a2 _ a is nilpotent. Then either a is nilpotent or for some polynomial q(x) with integer Received by the editors May 11, 1977 and, in revised form, July 20, 1977. AMS (MOS) subject classifications (1970). Primary 16A28. ? American Mathematical Society 1978
2 citations
TL;DR: In this article, it was shown that one can achieve the same result with two mappings satisfying eight necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup.
Abstract: In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.