Bands Of Semigroups Research Articles

Bands of λ-simple semigroups

Semigroups having a decomposition into a band of semigroups have been studied in many papers. In the present paper we give characterizations of various special types of bands of ?-semigroups and semilattices of matrices of ?- semigroups.

R-bands of r-archimedean ordered semigroups

Some semigroups (without order) are decomposable into r-archimedean semigroups. In the present paper we deal with the problem of decomposing certain ordered semigroups into r (l)-archimedean components. The semilattice congruences play an important role in the decomposition of semigroups -without order. When we pass from semigroups without order to ordered semigroups, the same role is played by the complete semilattice congruences. The characterization of complete semilattices of semigroups of a given type has been considered by the same authors. Band congruences play an important role in studying the decomposition of some ordered semigroups, like the decomposition of t-archimedean ordered semigroups. The r (l)-band congruences have been also proved to be useful in studying the decomposition of some types of ordered semigroups, especially the decomposition of r (l)-archimedean ordered semigroups. In this paper we first prove that an ordered semigroup S is an r (resp. l)-band of semigroups of a given type \( \mathcal{T} \) if and only if it is decomposable into pairwise disjoint subsemigroups Sα of S of type \( \mathcal{T} \) indexed by a band B such that SαSβ ⊆ Sαβ for all α, β ∈ B and Sα ∩ (Sβ] ≡ O implies α = βα (resp. α = αβ). Thenwe characterize the r (resp. l)-bands of r (resp. l)-archimedean semigroups.

Structure of radicals in bands of semigroups

It is well-known that semigroups of many important classes are decomposable into bands of their subsemigroups with more “rigid” structure. For efficient applications of radicals to semigroups of that kind, an information on the shape of radicals in bands of semigroups may be useful. Here we will investigate the Jacobson, Baer, Brown-McCoy and the least special radicals. The results were announced in [3]. The interaction of bands and radicals was also considered in [4] for the class of semigroups with zero; analogous problems for bands and radicals of associative rings were investigated in [5], [6] and [7].

Sturdy bands of semigroups

In this paper we consider band compositions determined by two transitive systems of monomorphisms, called the sturdy bands of semigroups.

Right zero unions of N-semigroups

=1 commutative semigroup is called archimedean if every element divides a positive integer power of each other element. An N-semigroup is one of four possible types of archimedean semigroups. Chrislock [l] extends the notion of archjmedeanness to noncommutative semigroups. An N*-semigroup is a noncommutative archimedean semigroup that can be thought of as a generalization of an N-semigroup. A basic reference on N-semigroups is [5]. Let F = {S a : u E R} be a disjoint family of semigroups. F has a right zero union, if there exists a semigroup T that is a disjoint union of the S, , where each S, is a left ideal of T. Thus, T is homomorphic onto the right zero semigroup R by mapping each S, to 01. It will be shown that an N*-semigroup is a right zero union of N-semigroups and conversely. Section 2 can be thought of as preliminary material. Section 3 examines right zero unions of N-semigroups. It will be shown in that section that a right zero union of N-semigroups is isomorphic to a subdirect product of an ,V-semigroup and a right zero semigroup. Conversely, any such subdirect product is a right zero union of N-semigroups. Section 4 proves, biter alia, the equivalence of the two concepts N*-semigroups and right zero unions of N-semigroups. An N-semigroup has a representation involving an abelian group G and a function I defined on G x G. Similarly, an N*-semigroup has a representation with an abelian right group H (if H = G x R then G is abelian) and an I* function defined on H x H. Section 4 will examine I* functions, and will conclude with a method for constructing them. Omitted proofs can be found in [4]. A g eneral reference to semigroup theory is [3]. The study of right zero unions is a special case of the study of “bands of semigroups.”

Bands of Semigroups

Bands of Semigroups

Bands of semigroups

In studying a general semigroup S, a natural thing to do is to decompose 5 (if possible) into the class sum of a set {Sa; aCl} of mutually disjoint subsemigroups Sa such that (1) each Sa belongs to some more or less restrictive type 13 of semigroup, and (2) the product SaSfi of any two of them is wholly contained in a third: SaSpClSy, for some yCI depending upon a and /3. We shall then say that 5 is a of semigroups of type 13. If, for every a and /3 in I, SaSp and SpSa are both contained in the same Sy, then we shall call S a semilattice of semigroups of type 13. We shall also be concerned with the following specialization of the notion of of semigroups. Suppose that / is the direct product JXK of two classes / and K. The subsemigroups Sa are then described by two subscripts: 5,-< (iCJ, kCK). Suppose moreover that SitSjxCSiK for all i, jCJ and all k, X£i£. We shall then call 5 a of semigroups of type 13. The primary purpose of the present paper is to show (Theorem 4) that a of semigroups of type 13 is a semilattice of semigroups each of which is a of semigroups of type 13. The rest of the paper is devoted to giving necessary and sufficient conditions on a semigroup 5 that it be a or a semilattice of (1) simple semigroups, (2) completely simple semigroups, and (3) groups. (Throughout this paper we use the term simple to mean simple without zero, i.e. a simple semigroup is containing no proper two-sided ideal whatever.) For (1), we have the elegant condition, aCSa2S for all aCS, due to1 Olaf Andersen [l ]. If a semigroup 5 is a class sum of [completely] simple semigroups, it is also a semilattice of [completely] simple semigroups. But a class sum of groups need not be a of groups, nor need a of groups be a semilattice of groups; these three categories are characterized by Theorems 6, 7, and 8, respectively. We note that a semigroup 5 is a band of groups of order one if and only if each element of S1 is idempotent. In this case we call 5 simply a band, and consequently make the definition: a is a semigroup every element of which is idempotent. By the same token, we define a semilattice to be a commutative band. A matrix of groups