Showing papers on "Bicyclic semigroup published in 1972"
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TL;DR: In this article, the authors extend the usual ring concept to the case where the underlying group and semigroup are respectively ann-ary group and anm-ary semigroup, and they apply it to homomorphism theorems.
Abstract: The following paper is concerned with the extension of the usual ring concept to the case where the underlying group and semigroup are respectively ann-ary group and anm-ary semigroup. Most attention has been paid to homomorphism theorems and to some idealtheoretic aspects.
45 citations
30 citations
TL;DR: In this article, the authors give a complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semi-group and its congruences.
Abstract: This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences. 0. Let G be a subgroup of the multiplicative group of positive real numbers and let P denote the subsemigroup of G consisting of all x E G with x ? 1. Denote by Wp the class of all inverse semigroups H for which there isa homomorphism f: P -> H such that f(P) generates H (no proper inverse subsemigroup of H contains f(P)). We shall call such semigroups H one-parameter inverse semigroups and denote by W= UP Wp the class of all one-parameter inverse semigroups. The class W contains well-known semigroups. For example, each homomorphic image of a subgroup of R, the positive real numbers, is a member of W. Also the bicyclic semigroup B is a member of W, as is seen by noting that B is generated by a copy of the nonnegative integers. Indeed, if H is any elementary inverse semigroup, then H1 is generated by a homomorphic image of the nonnegative integers, and so is a one-parameter inverse semigroup. The main purpose of this paper is to describe all one-parameter inverse semigroups. In the process of doing this, we shall construct what we term the free oneparameter inverse semigroups Fp, one for each subgroup G of R and its associated semigroup P. The semigroup Fp is the only inverse semigroup (up to isomorphism) generated by a subsemigroup isomorphic with P which has the property that each homomorphism f: P -> S, an inverse semigroup, extends uniquely to a homomorphism f: Fp -> S. In particular, every H E Wp is a homomorphic image of Fp. We thus adopt the point of view that by describing Fp and the lattice of congruences of Fp for arbitrary P, we will have described all one-parameter inverse semigroups. We shall assume a certain familiarity with the algebraic theory of semigroups, particularly inverse semigroups. (See Clifford and Preston [1].) The existence and uniqueness of Fp is a consequence of a theorem due to McAlister [3, Theorem 33]. We were greatly aided in the actual description of Fp Presented to the Society, August 27, 1969; received by the editors June 10, 1969. AMS 1970 subject classifications. Primary 20M10; Secondary 20M05, 22A15.
25 citations
01 Jan 1972
TL;DR: In this article, an upper bound for the global dimension of the semigroup algebra of a finite regular monoid in terms of an ideal series for the monoid is determined by the length of the longest chain of f-classes of a monoid.
Abstract: An upper bound for the global dimension of the semigroup algebra of a finite regular monoid in terms of an ideal series for the monoid is determined by the partially ordered set of f-classes of the monoid. In particular, if the monoid is combinatorial, the global dimension of the algebra is bounded by the sum of the global dimension of the coefficient ring and twice the length of the longest chain of f-classes in the monoid. In [3] we considered a finite regular monoid S and a composition series of ideals S=I1DI2.2=.I.. For a commutative ring with identity k we then defined a(Ij)=O, 1, or 2, respectively, depending on whether the algebra kIj/kIj+l has a two-sided identity, has no two-sided identity but has a right or left identity, or has neither right nor left identity. Then we set ,u(S)=a(I-) )+c()+2).. +v(I ) and found an upper bound for l.gl.dim kS which involved ,u(S). In particular, if S is combinatorial (i.e., each subgroup is trivial), or if kG is semisimple for each subgroup G of S, then l.gl.dim kS<,(S)+l.gl.dim k. The purpose of this paper is to show that an estimate -(S) can be computed as above using an ideal series for S which is in general shorter than a composition series. For semigroup terminology we follow [1] or [2]. We form an ideal series S=IDJ)1' * *-Im as follows: Let IP be the unique minimal ideal of S, and for r=m-1, ,1 let Jr be such that irl/lJ+ is the union of all 0-minimal ideals in S/Ir;+. In this ideal series m is the length of the longest chain in the partially ordered set of f-classes of S. In particular, m=n if and only if S is a chain of f-classes; otherwise m
20 citations
TL;DR: In this article, the structure of regular semigroups without contractions is studied, whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idems in different!D-classes, or all of whose ideals are categorical or satisfy a variation thereof.
Abstract: The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different !D-classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of Dmajorization of idempotents, where D is a ED-class, !-majorization, E-categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.
18 citations
TL;DR: In this article, a bisimple left inverse semigroup S with identity element e as a quotient of the cartesian product Le×Re of the L-class Le of S and the R-class Re of S containing e is described.
Abstract: A semigroup S is called left inverse if every principal right ideal of S has a unique idempotent generator. We investigate D-classes of regular semigroups and of left inverse semigroups, characterizing those which are subsemigroups. We give a description of a bisimple left inverse semigroup S with identity element e as a quotient of the cartesian product Le×Re of the L-class Le of S and the R-class Re of S containing e. We also describe the maximal inverse semigroup homomorphic image of S.
11 citations
TL;DR: A solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock's method as part of their construction is given in this article. But it is not a complete solution.
Abstract: D. R. Brown and M. Friedberg have conjectured that each compact abelian semigroup can be embedded in a compact divisible semigroup. V. R. Hancock proved that each abelian algebraic semigroup can be embedded in a divisible abelian algebraic semigroup. In this paper we provide a partial solution to the conjecture of Brown and Friedberg by employing a topological version of Hancock's method as part of our construction. A theorem giving sufficient conditions for the Bohr compactification of weakly reductive semigroups to be injective is proved and used in the proof of our main result.
8 citations
TL;DR: This paper is concerned with the structure of the semigroup of a linear nonsingular automaton and gives necessary and sufficient conditions for any semigroup to be isomorphic with a linear non-singular subautomaton.
Abstract: This paper is concerned with the structure of the semigroup of a linear nonsingular automaton and gives necessary and sufficient conditions for any semigroup to be isomorphic with the semigroup of a linear non-singular subautomaton.
7 citations
6 citations
TL;DR: A condition under which the factors of a subset of a free monoid are unique is given and a condition is given under which a subsets of afree monoid may have factors.
Abstract: We give a condition under which the factors of a subset of a free monoid are unique. Some related results are demonstrated and a condition is given under which a subset of a free monoid may have factors.
4 citations
TL;DR: The semigroup of quotients Q corresponding to an arbitrary right quotient filter on a semigroup S is also a semilattice of groups as mentioned in this paper, which is a regular semigroup in which all idempotents are central.
Abstract: Let S be a semigroup with zero which is a semilattice of groups. In [6], McMorris showed that the semigroup of quotients Q=Q(S) corresponding to the filter of “dense” right ideals of the semigroup S is also a semilattice of groups. He accomplished this by noting that Q is a regular semigroup in which all idempotents are central, an equivalent formulation of a semilattice of groups. In this paper we develop the semigroup of quotients Q corresponding to an arbitrary right quotient filter on S (as defined herein) and note the above result in this more general setting by explicitly constructing a semigroup which is isomorphic to Q. We also see that the underlying semilattice for Q in this case is isomorphic to a semigroup of quotients of the original semilattice for the semigroup S.
TL;DR: In particular, the authors showed that any semigroup can be explicitly rebuilt from its quotient by any congruence contained in ℋ and certain groups and mappings, which underlies most of the classical and recent structure theorems in semigroup theory.
Abstract: Any semigroup can be quite explicitly rebuilt from its quotient by any congruence contained in ℋ and certain groups and mappings. This fact underlies most of the classical and recent structure theorems in semigroup theory, and yields new ones, e.g. concerning finite commutative semigroups.
TL;DR: In this paper, it was shown that every congruence on a generalized inverse semigroup is uniquely determined by a congruense on its associated inverse semiigroup, left normal band, and right normal band.
Abstract: A generalized inverse semigroup is a regular semigroup whose idempotents satisfy a permutation identity X1 X2...Xn=Xp1 Xp2...Xpn, where (P1, P2..., Pn) is a nontrivial permutation of (1, 2,..., n). Yamada [4] has given a complete classification of generalized inverse semigroups in terms of inverse semigroups, left normal bands, and right normal bands. In this paper we show that every congruence on a generalized inverse semigroup is uniquely determined by a congruence on its associated inverse semigroup, left normal band, and right normal band. A converse is also provided.
TL;DR: In this article, it was shown that the Green relations in the endomorphism semigroup, End R, for a ring R in a given class, are restrictions of those in the transformation semigroup.
Abstract: In this paper we investigate semigroups of ring endomorphisms of several classes of rings. As one result we find that the Green relations in the endomorphism semigroup, End R, for a ring R in a given class, are restrictions of those in the transformation semigroup
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TL;DR: The theory of ideal extensions of semigroups is dependent upon adequate knowledge of the translational hull of a semigroup as discussed by the authors, and it is shown that the semilattice of idempotents of an arbitrary inverse semigroup S and the group of units of S can be characterized in terms of this structure.
Abstract: The theory of ideal extensions of semigroups is dependent upon adequate knowledge of the translational hull. We characterize here certain basic structures in the translational hull, Ω(S), of an arbitrary inverse semigroup S: in particular, the semilattice of idempotents of Ω(S) and the group of units of Ω(S). An isomorphic copy of each of these structures is given in terms of the semilattice of idempotents of S.
TL;DR: In this paper, all finite commutative subdirectly irreducible semigroups of this type are realized as permutation group extensions of nilpotent semiigroups.
Abstract: This paper is devoted to completing the solution to the problem of constructing all finite commutative subdirectly irreducible semigroups. Those semigroups of this type which were formerly unknown are realized as certain permutation group extensions of nilpotent semigroups. The results in this paper extend the efforts in this area by G. Thierrin and B. M. Schein.
TL;DR: In this paper, the problem of extending a primitive inverse semigroup by another can be reduced to that of extending one Brandt semigroup with only a finite number of idempotents.
Abstract: Every inverse semigroup containing a primitive idempotent is an ideal extension of a primitive inverse semigroup by another inverse semigroup. Consequently, in developing the theory of inverse semigroups, it is natural to study ideal extensions of primitive inverse semigroups (cf. [3; 7]). Since the structure of any primitive inverse semigroup is known, an obvious type of ideal extension to consider is that of one primitive inverse semigroup by another. In this paper, we will construct all such extensions and give an abstract characterization of the resulting semigroup. The problem of extending one primitive inverse semigroup by another can be essentially reduced to that of extending one Brandt semigroup by another Brandt semigroup. The latter problem has been solved by Lallement and Petrich in [3] in case the first Brandt semigroup has only a finite number of idempotents.
TL;DR: A semigroup is a semigroup that is completely [O-] simple iff it is isomorphic to a binary Boolean (I×I)-matrices such that for every p,q∈I there exists precisely one matrix (aij)∈ϕ such that apg=1.
Abstract: A semigroup is completely [O-] simple iff it is isomorphic to a semigroup ϕ of binary Boolean (I×I)-matrices (where I is a set) such that for every p,q∈I there exists precisely one matrix (aij)∈ϕ such that apg=1. Every such isomorphic representation of a completely [O-] simple semigroup is an arbitrary inflation of a reduced representation and all reduced representations are equivalent.
TL;DR: In this article, the structure semigroup of the L 1 -algebra of an Arens-Singer semigroup is discussed and a complete and rather unexpected description of a nontrivial structure semiigroup is provided.
Abstract: In this paper we discuss the structure semigroup of the L 1 -algebra of an Arens-Singer semigroup. Arising from this study we provide a complete and rather unexpected description of a nontrivial structure semigroup. We then link the above ideas with that of the almost periodic compactification of the semigroup.
TL;DR: In this article, it was shown that any inverse semigroup is isomorphic to an inverse subsemigroup of Jx for some semilattice X, where X is a partially ordered set.
Abstract: If S is an inverse semigroup and 0 is the relation on the lattice A(S) of congruences on S defined by saying that two congruences ρl9p2 are ^-equivalent if and only if they induce the same partition of the idempotents then Θ is a congruence on Λ(S) and each #-class is a complete modular sublattice of A(S). If X is a partially ordered set then Jx denotes the inverse semigroup of one-to-one partial transformations of X which are order isomorphisms of ideals of X onto ideals of X, while if X is a semilattice, Tz denotes the inverse subsemigroup of Jx consisting of those elements a whose domain Δ(a) and range f (a) are principal ideals. It is shown that any inverse semigroup is isomorphic to an inverse subsemigroup of Jx for some semilattice X.
TL;DR: In this paper, the authors characterized the finite nilpotent semigroups by the length of the ideal lattice of ideals of a null semigroup and generalized a theorem of T. Tamura and M. Yamada.
Abstract: In this paper, we establish several properties of the lattice of ideals of a nilpotent semigroup. In particular, we characterize the finite nilpotent semigroups by the length of this lattice, and we generalize a theorem of T. Tamura and M. Yamada [5], concerning the ideal extension of a nilpotent semigroup by a null semigroup.