Showing papers on "Cancellative semigroup published in 1989"
TL;DR: The size-time complexity of computing prefixes with Boolean networks, which are synchronized interconnections of Boolean gates and one-bit storage devices, is characterized and area-time optimal circuits are obtained for both boundary and nonboundary I/O protocols.
Abstract: The prefix problem consists of computing all the products x0x1 … xj (j = 0, … , N - 1), given a sequence x = (x0, x1, … , xN-1) of elements in a semigroup. In this paper we completely characterize the size-time complexity of computing prefixes with Boolean networks, which are synchronized interconnections of Boolean gates and one-bit storage devices. This complexity crucially depends upon two properties of the underlying semigroup, which we call cycle-freedom (no cycle of length greater than one in the Cayley graph of the semigroup), and memory-induciveness (arbitrarily long products of semigroup elements are true functions of all their factors). A nontrivial characterization is given of non-memory-inducive semigroups as those whose recurrent subsemigroup (formed by the elements with self-loops in the Cayley graph) is the direct product of a left-zero semigroup and a right-zero semigroup. Denoting by S and T size and computation time, respectively, we have S = T((N/T)log(N/T)) for memory-inducive non-cycle-free semigroups, and S = T(N/T) for all other semigroups. We have T e [O(log N), O(N)] for all semigroups, with the exception of those whose recurrent subsemigroup is a right-zero semigroup, for which T e [O(1), O(N)]. The preceding results are also extended to the VLSI model of computation. Area-time optimal circuits are obtained for both boundary and nonboundary I/O protocols.
16 citations
TL;DR: In this paper, the authors studied the question of whether a cancellative semigroup can be embedded into a group if it satisfies Ore's condition of being either left or right reversible.
Abstract: It is well known that a cancellative semigroup can be embedded into a group if it satisfies “Ore’s condition” of being either left or right reversible. However Ore’s condition is by no means necessary, so it is natural to ask which subsemigroups of a group are left or right reversible, or satisfy a condition of a similar type. In the present paper we study this question on open subsemigroups of connected locally compact groups; we also show how to use concepts related with reversibility to prove assertions like the following: Suppose thatS is an open subsemigroup of a connected Lie groupG such that 1 ∈\(\bar S\). IfG is solvable or ifS is invariant thenS is connected andS determinesG uniquely; that is to say, ifS can be embedded as an open subsemigroup into a connected Lie groupG’ thenG’ is isomorphic withG. Examples show that there are non-connected open subsemigroupsS of Sl(2,R) with 1 ∈\(\bar S\) and such that the uniqueness assertion fails.
15 citations
TL;DR: For a semigroup S, the set of all subsemigroups of S × S, with the operations of composition and involution and the relation of set-theoretic inclusion, forms the bundle of correspondences of S, denoted by bS as discussed by the authors.
9 citations
01 Jan 1989
6 citations
19 Jun 1989
TL;DR: In this article, the authors give an example of a positive semigroup on a Banach lattice whose semigroup dual is not a BLS. And they show that a positive semi-group on a BSL can be represented by a positive non-convex semigroup.
Abstract: We give an example of a positive semigroup on a Banach lattice whose semigroup dual is not a Banach lattice.
3 citations
01 Apr 1989
TL;DR: In this article, the existence of almost translation invariant ultrafilters on any cancellative semigroup using the ultrafilter approach was proved by Glazer and Glazer this article.
Abstract: We present a short ultrafilter proof of a result which has applications in combinatorial number theory and which has previously relied on the theory of compact semigroups. The existence of non-fixed ultrafilters p on N such that whenever A E p one has {x e N: A x e p} e p, called almost translation invariant by Galvin, has always been of interest because it is closely related to the validity of GrahamRothschild conjecture. In 1974, the conjecture was proved by Hindman [5] in ZFC. Combined with the proof in [4] which used the continuum hypothesis this yielded a CH proof of the existence of almost translation invariant ultrafilters. In 1975, Glazer proved their existence without using the continuum hypothesis [2]. His approach was to define an addition on ultrafilters on a semigroup S, so that the almost translation invariant ultrafilters become idempotents and then use Ellis' theorem (Lemma 2.9 of [3]) about the existence of idempotents in compact right topological semigroups. In this paper, we prove directly the existence of such ultrafilters on any cancellative semigroup using ultrafilter approach only. 1. Definition. Let (S, +) be a semigroup, not necessarily commutative, and let X, X' be filters on S. For any A C S we define Q,(A) = {x E S: A x ez},v _+ = {A C S: Q9(A) Es}, where A-x=f{yeS:y+xeA}. The symbol + is just a notion for the above described operation on filters. The fact that v + 5 is a filter follows from steps (i) and (iii) of the following lemma. (In [1], Lemma 5.15 shows that v +X is a filter on S and that + is an associative operation on the set of filters. In the following discussion the associativity of + is not needed.) All parts of the lemma are easy to prove. Received by the editors February 7, 1989 and, in revised form, February 21, 1989. 1980 Mathemnatics Subject Classification (1985 Revision). Primary 54A25. KeY words and phrases. Ultrafilters. (? 1989 American Mathematical Society 0002-9939/89 $1.00 + S.25 per page
3 citations
Journal Article•
2 citations
TL;DR: In this article, a sufficient condition for a fixed operator in the algebra A to be embeddable in a C0 semigroup of operators in A is given, under mild conditions.
2 citations
TL;DR: In this article, the authors considered right invariant right holoids, which are semigroups with left cancellation rule, ordered by right divisibility, and were able to give a structure theorem under weak additional assumptions.
Abstract: Through this paper we are concerned with right invariant right holoids which are semigroups with left cancellation rule, ordered by right divisibility. We are able to give a structure theorem under weak additional assumptions. In the proof we use factorsystems which occur in homological algebra.