Showing papers on "Cancellative semigroup published in 1975"
68 citations
TL;DR: In this article, a cohomological framework for inverse semigroups is provided, which will not only fit the extension problem, but also discuss some apparently new notions such as complementation and inner automorphism for inverse semiigroups.
58 citations
TL;DR: The strong Burnside problem has an affirmative answer for semigroups of finite dimensional matrices over a skew field as discussed by the authors, and it follows that a torsion semigroup embeddable in the multiplicative semigroup of an algebra over a field satisfying a polynomial identity is locally finite.
55 citations
27 citations
25 citations
TL;DR: For a 2-sided fir ring with no non-trivial right invariant elements, the authors showed that the non-zero T-ideals of such an algebra R form a free subsemigroup of the semigroup of all nonzero ideals, and thus the induced semigroup structure on V is also free.
Abstract: If R is a 2-sided fir (free ideal ring) with no non-trivial right invariant elements, we shall find that the non-zero 2-sided ideals of R, under the usual multiplication of ideals, form a free semigroup with 1. In particular, this holds when R is a free associative algebra over a field. (We also consider the operations of multiplying right ideals by 2-sided ideals to get right ideals, 2-sided ideals by left ideals to get left ideals, and right ideals by left ideals to get additive subsets, and find that these actions too are free, in the appropriate senses.) It is well known that the set V of proper subvarieties of the variety of associative algebras over a field K is in natural correspondence with the set T of non-zero T-ideals (completely invariant ideals) of any free /sf-algebra R = K(X} on an infinite set of indeterminates. A product of T-ideals is again a T-ideal, and we shall show that the non-zero T-ideals of such an algebra R form a free subsemigroup of the semigroup of all non-zero ideals. Thus the induced semigroup structure on V is also free. Finally, we show that in any 2-fir R without non-trivial invariant elements, a 2-sided ideal / is uniquely determined by its (/?, /?)-bimodule structure. (This result is independent of the others.)
22 citations
TL;DR: In a series of papers ((1967, (1967a) and (1967b)) Magill has considered the semigroups J ( X, Y ; θ ) (definition below), a natural, but extensive, generalization of the usual transformation semigroup J( X ).
Abstract: In a series of papers ((1967), (1967a) and (1967b)) Magill has considered the semigroups J ( X, Y ; θ ) (definition below), a natural, but extensive, generalization of the usual transformation semigroup J ( X ). They have also been studiedin Sullivan (to appear). Under the assumption that θ be onto Magill described their automorphisms and determined when one J ( X, Y ; θ ) is isomorphic to another.
22 citations
12 citations
TL;DR: In this paper, Clifford et al. provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-side congruences from its kernel.
Abstract: By the kernel of a one-sided (left or right) congruence p on an inverse semigroup S, we mean the set of p-classes which contain idempotents of S. We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup S which are induced by a one-sided congruence on S and provide a characterization of the maximum and minimum one-sided congruences on S inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section. 1. Basic notions, terminology. We adhere throughout to the notation and terminology of A. H. Clifford and G. B. Preston [1]. Throughout the paper, S will always denote an inverse semigroup (i.e., for each a ES, there exists a unique element a~l E S suchthat a = aa~la and a-1 = a~1aa~1) and Es will denote the set of idempotents of S. The elementary properties of inverse semigroups may be found in [1]. In particular, we shall liberally use, without comment, the fact that Es is a semilattice (a commutative semigroup of idempotents) and that a~lEsa CES Va E S. We shall also use the fact that if S is an inverse semigroup then the Green's relations L and R on S are given by L = {(a, b) G S x S: a~xa = b~lb}
12 citations
TL;DR: In this article, it was shown that a semigroup of the type being considered in this paper is isomorphic to a Rees matrix semigroup over a commutative cancellative semigroup.
Abstract: In this paper it is shown that each semigroup which is a matrix of commutative cancellative semigroups has a "quotient semigroup" which is a completely simple semigroup with abelian maximal subgroups. This result is proved by explicitly constructing the quotient semigroup. The paper also gives necessary and sufficient conditions for a semigroup of the type being considered in the paper to be isomorphic to a Rees matrix semigroup over a commutative cancellative semigroup. Several special cases and examples are also briefly discussed. The study of the semigroups in the title was initiated by Petrich [7] in connection with commutative separative semigroups. It was conjectured in that paper that matrices of commutative cancellative semigroups can be embedded into Rees matrix semigroups over abelian groups. This paper answers the conjecture affirmatively. We also study the embedding and use it to characterize several special cases of matrices of commutative cancellative semigroups. 0. Preliminaries and summary. We use S to represent a semigroup. If there is a congruen Ce p on c for which S/p is a rectangular band whose classes are all commutative cancellative semigroups, then we say S is a matrix of commutative cancellative semigroups. Since a rectangular band may be considered as I x A, the product of a left and right zero semigroup respectively, we will write S = IUA Si A for a matrix of commutative cancellative semigroups, whose classes are the Six. In case the rectangular Received by the editors March 22, 1974. AMS (MOS) subject classifications (1970). Primary 20M10, 20M30.
7 citations
TL;DR: In this article, an application of radical theory to the structure of commutative semigroups via their semilattice decomposition is presented, where maximal group congruences and semisimplicity are characterized for certain classes of commuttative semigen and N-seinigroups.
Abstract: This paper presents an application of radical theory to the structure of commutative semigroups via their semilattice decomposition. Maximal group congruences and semisimplicity are characterized for certain classes of commutative semigroups and N-seinigroups. The concepts of a radical theory and semisimplicity in semigroups analogous to that of ring theory have been studied by a number of authors, both as a general theory, and applied to specific classes of semigroups. (See e.g. [1], [3], [5]-[9].) In this paper we apply these techniques to a study of the structure of commutative semigroups. Every semigroup S has a least congruence it such that Slit is a semilattice Y. Each congruence class of ,u is a subsemigroup of S, and the collection .Sa, a E Y, of congruence classes is called the greatest semi. lattice decomposition of S. Conversely, if Y is a semilattice and ISa} a E Y, is a collection of pairwise disjoint semigroups, then any semigroup S U Its : a E Y} with the property that S S C S for a, f E Y is a semilattice composition of the Sa. Thus a natural approach to the structure of a given type of semigroup is through a characterization of the semilattice decompositions, the structure of the Sa, and the semilattice compositions. The semilattice decomposition of commutative semigroups was described by Tamura and Kimura [12]. In this case the semigroups Sa are the maximal archimedean subsemigroups, where Sa archimedean means given any two elements of Sa each divides some power of the other. Conversely, the general solution of the semilattice composition problem is known (see [10, Theorem III.7.2]). If r is a congruence on a semigroup S, we say r is modular if there exists an element e in S such that (ex)r(xe)rx for all x in S. The radical Received by the editors June 28, 1974. AMS (MOS) subject classifications (1970). Primary 20M10.
TL;DR: Cancellative, idempotent-free semigroups with non-empty center are characterized in terms of a Schreier extension of Cancellative P pivoted Semigroups as discussed by the authors.
Abstract: Cancellative, idempotent-free semigroups having non-empty center are characterized in terms of a Schreier extension Cancellative pivoted semigroups with non-empty center are characterized as a group or in terms of a triple (G,H,I), where G is a group, H is either empty or a subgroup of G and I is a function mapping GxG into the non negative integers
01 Jan 1975
TL;DR: In this article, it was shown that a probability measure is defined to be quasi-nilpotent if and only if its support is quasinilpotent (i.e., the probability measure converges to the Dirac measure at 0).
Abstract: In a compact semigroup S with zero 0, a subset A of S is called quasi-nilpotent if the closed semigroup generated by A contains 0. A probability measure A on S is called nilpotent if the sequence (u') converges to the Dirac measure at 0. It is shown that a probability measure is nilpotent if and only if its support is quasi-nilpotent. Consequently, the set of all nilpotent measures on S is convex and everywhere dense in the set of all probability measures on S and the union of their supports is S. In a topological semigroup with zero 0, an element x is termed nilpotent if xn-* 0 as n --[5]. This definition has an obvious extension to subsets of the semigroup, i.e. a subset A is nilpotent if A-* o as n *-. Now we call a subset B of the semigroup quasi-nilpotent if the closed semigroup generated by B contains the ze-o 0. It is shown that, when the topological semigroup is compact, a singleton is nilpotent if and only if quasinilpotent. Then we investigate the set of probability measures on a compact semigroup and characterize a nilpotent probability measure as a measure with quasi-nilpotent support. Let S be a topological semigroup with zero 0, and A a subset of S. Let S(A) denote the semigroup generated by A, i.e. S(A) = Un=An. It is trivial that any subset containing 0 is quasi-nilpotent; in particular, the set N(S) of nilpotent elements of S is quasi-nilpotent. From the semigroup S given in Example 6 below, in which N(S) = [0, 1) and N(S)n = N(S) for all n [4, p. 56], we see that N(S) is not nilpotent. Theorem 1. Let A be a subset of S. Then (i) If S(A)n N(S) 0 (where the bar denotes closure), then A is quasi-nilpotent. (ii) If An is quasi-nilpotent for some n, then A itself is quasi-nilpotent. Proof. (i) Take a E S(A)n N(S). In view of the fact that an 0 O, we have 0 E S(A), i.e. A is quasi-nilpotent. (ii) Since S(An) C S(A) and 0 E S(An), it follows that 0 E S(A), and the theorem is proved. We remark that, if An is nilpotent for some n, then A is also nilpotent, by a similar argument to that given in the proof of Lemma 2.1.4 of [4]. Received by the editors April 3, 1974 and, in revised form, July 15, 1974. AMS (MOS) subject classifications (1970). Primary 22A20, 43A05, 60B15; Secondary 22A15.
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.
TL;DR: In this article, it was shown that for any orthomodular poset P, there exists a partial Baer *-semigroup whose closed projections are order isomorphic to P preserving ortho-complementation.
Abstract: It is shown in Gudder and Schelp (1970) that partial Baer *-semigroups coordinatize orthomodular partially ordered sets (orthomodular posets). This means for P an orthomodular poset there exists a partial Baer *-semigroup whose closed projections are order isomorphic to P preserving ortho-complementation. This coordinatization theorem generalizes Foulis (1960) in which orthomodular lattices are coordinatized by Baer *-semigroups. In particular Foulis (unpublished) shows that any complete atomic Boolean lattice is coordinatized by a Bear *-semigroup of relations. Since Greechie (1968), (1971) shows that a whole class of orthomodular posets can be formed by “pasting” together Boolean lattices, it is natural to consider the following problem. Let y be a family of Baer *-semigroups of relations which coordinatize the family B of complete atomic lattices. Is it possible to construct a partial *-semigroup of relations R which contains each member of Y such that when P is an orthomodular poset obtained by a “Greechie pasting” of members of 38 then 91 coordinatizes R This question is considered in the sequel and answered affirmatively for a certain subclass of “Greechie pasted” orthomodular posets. In addition the construction of 8)t nicely fulfills another objective in that it provides us with “nontrivial” coordinate partial Baer *-semigroups for a whole family of well known orthomodular posets. This is particularly significant since the only other known coordinate partial Baer *-semigroups, for those posets in this family which are not lattices, are the “minimal” ones given in Gudderand and Schelp (1970).