Commutative semigroups whose homomorphic images in groups are groups

In this paper we study commutative semigroups whose every homomorphic image in a group is a group. We find that for a commutative semigroup S, this property is equivalent to S being a union of subsemigroups each of which either has a kernel or else is isomorphic to one of a sequence T₀, T₁, T₂, ... of explicitly given, countably infinite semigroups without idempotents. Moreover, if S is also finitely generated then this property is equivalent to S having a kernel.     

Lower-Modular Elements of the Lattice of Semigroup Varieties

We call a semigroup variety modular [upper-modular, lower-modular, neutral] if it is a modular [respectively upper-modular, lower-modular, neutral] element of the lattice of all semigroup varieties. It is proved that if V is a lower-modular variety then either V coincides with the variety of all semigroups or V is periodic and the greatest nil-subvariety of V may be given by 0-reduced identities only. We completely determine all commutative lower-modular varieties. In particular, it turns out that a commutative variety is lower-modular if and only if it is neutral. A number of corollaries of these results are obtained.     

Finite union of commutative power joined semigroups

A commutative semigroup is called power joined if for every element a, b there are positive integers m, n such that a^m=bⁿ. A commutative power joined semigroup is archimedean (p. 131, [3]) and cannot be decomposed into the disjoint union of more than one subsemigroup. Every commutative semigroup is uniquely decomposed into the disjoint union of power joined subsemigroups which are called the power joined components. This paper determines the structure of commutative archimedean semigroups which have a finite number of power joined components. The number of power joined components of commutative archimedean semigroups is one or three or infinity. The research for this paper was supported in part by NSF Grant GP-11964.     

On the Finite and Non-finite Generation of Finitary Power Semigroups

The finitary power semigroup of a semigroup S, denoted P_f(S), is the set of finite subsets of S with respect to the usual set multiplication. Semigroups with finitely generated finitary power semigroups are characterised in terms of three other properties. From this statement there are drawn several corollaries. It follows that P_f(S) is not finitely generated if S is infinite and in any of the following classes: commutative; Bruck-Reilly extensions; inverse semigroups that contain an infinite group; completely zero-simple; completely regular.     

Power Centralized Semigroups

A semigroup is said to be power centralized if for every pair of elements x and y there exists a power of x commuting with y. The structure of power centralized groups and semigroups is investigated. In particular, we characterize 0-simple power centralized semigroups and describe subdirectly irreducible power centralized semigroups. Connections between periodic semigroups with central idempotents and periodic power commutative semigroups are discussed.     

Semigroups of partial isometries

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.     

Cancellative Orders

left order in Q and Q is a semigroup of left quotients of S if every q∈Q can be written as q=a^*b for some a, b∈S where a^* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. Perhaps surprisingly, a semigroup, even a commutative cancellative semigroup, can have non-isomorphic semigroups of left quotients. We show that if S is a cancellative left order in Q then Q is completely regular and the {\cal D}-classes of Q are left groups. The semigroup S is right reversible and its group of left quotients is the minimum semigroup of left quotients of S. The authors are grateful to the ARC for its generous financial support.     

Idempotent bounded C-semigroups

A semigroup with zero isidempotent bounded (IB) if it is the 0-direct union of idempotent generated principal left ideals and the 0-direct union of idempotent generated principal right ideals. Notable examples are completely 0-simple semigroups and the wider class of primitive abundant semigroups. Significant to the structure of these semigroups is that they are all categorical at zero. In this paper we describe IB semigroups that are categorical at zero in terms ofdouble blocked Rees matrix semigroups. This generalises Fountain's characterisation of primitive abundant semigroups via blocked Rees matrix semigroups [1], which in turn yields the Rees theorem for completely 0-simple semigroups.     

A generalized Clifford theorem of semigroups

A U-abundant semigroup S in which every H-class of S contains an element in the set of projections U of S is said to be a U-superabundant semigroup.This is an analogue of regular semigroups which are unions of groups and an analogue of abundant semigroups which are superabundant.In 1941,Clifford proved that a semigroup is a union of groups if and only if it is a semilattice of completely simple semigroups.Several years later,Fountain generalized this result to the class of superabundant semigroups.In this p...     

Semigroups with Boolean congruence lattices

A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, respectively, with Boolean (...) congruence lattice are completely determined.     

Presentations for subsemigroups of finitely generated commutative semigroups

We introduce the concept of presentation for subsemigroups of finitely generated commutative semigroups, which extends the concept of presentation for finitely generated commutative semigroups. We show that for every subsemigroup of a finitely generated commutative semigroup there are special presentations which solve the word problem in the given subsemigroup. Some properties like being cancellative, reduced and/or torsion free are studied under this new point of view. This paper was supported by the project DGES PB96-1424.     

Note on exclusive semigroups

In the SEMIGROUP FORUM, Vol. 1, No. 1, B. M. Schein proposed the following problem: Describe the structure of semigroups S such that for every a,b,c∈S, abc=ab, bc or ac. At present, we shall call such a semigroup S anexclusive semigroup. Recently, the author heard that the structure of commutative exclusive semigroups was completely determined by T. Tamura [3]. In this paper, we deal with exclusive semigroups which are not necessarily commutative. The paper is divided into three sections. At first, the structure of exclusive semigroups whose idempotents form a rectangular band will be clarified. Next, we shall investigate a certain class of exclusive semigroups called “exclusive homobands”. Especially, in the final section we shall deal with medial exclusive homobands and show how to construct them. The proofs are omitted and will be given in detail elsewhere.     

On the existence of equivalent finite invariant measures

Summary Necessary and sufficient conditions are given for the existence of a finite measure which is equivalent to a given measure and invariant with respect to each transformation in a given commutative semigroup of measurable null-invariant point transformations. This result was already known for denumerably generated semigroups. A complementary result is proved which states that if one such equivalent measure exists, then there exists a unique equivalent measure which agrees with the original measure on the invariant sets.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.     

Orthodox semigroups whose idempotents satisfy a certain identity

An orthodox semigroup S is called a left [right] inverse semigroup if the set of idempotents of S satisfies the identity xyx=xy [xyx=yx]. Bisimple left [right] inverse semigroups have been studied by Venkatesan [6]. In this paper, we clarify the structure of general left [right] inverse semigroups. Further, we also investigate the structure of orthodox semigroups whose idempotents satisfy the identity xyxzx=xyzx. In particular, it is shown that the set of idempotents of an orthodox semigroup S satisfies xyxzx=xyzx if and only if S is isomorphic to a subdirect product of a left inverse semigroup and a right inverse semigroup.     

完全■-单半群

完全■-单半群是完全单半群和完全■~*-单半群在U-半富足半群类中的一个自然推广.本文证明了半群S是完全■-单半群,当且仅当S同构于幺半群T上的正规Rees矩阵半群■(T;I,A;P).这一结果不仅推广了完全单半群的著名Rees定理,而且推广了任学明和岑嘉评在2004年建立的完全■~*-单半群的一个结构定理.     

Permanent Semigroups

A permanent semigroup is a semigroup of n × n matrices on which the permanent function is multiplicative. If the underlying ring is an infinite integral domain with characteristic p > n or characteristic 0 we prove that any permanent semigroup consists of matrices with at most one nonzero diagonal. The same result holds if the ring is a finite field with characteristic p > n and at least n²+n elements. We also consider the Kronecker product of permanent semigroups and show that the Kronecker product of permanent semigroups is a permanent semigroup if and only if the pennanental analogue of the formula for the determinant of a Kronecker product of two matrices holds. This latter result holds even when the matrix entries are from a commutative ring with unity.     

Varieties and Nilpotent Extensions of Permutative Periodic Semigroups

We investigate certain semigroup varieties formed by nilpotent extensions of orthodox normal bands of commutative periodic groups. Such semigroups are shown to be both structurally periodic and structurally commutative, and are therefore structurally inverse semigroups. Such semigroups are also shown to be dense semilattices of structurally group semigroups. Making use of these structure decompositions, we prove that the subvariety lattice of any variety comprised of such semigroups is isomorphic to the direct product of the following three sublattices: its sublattice of all structurally trivial semigroup varieties, its sublattice of all semilattice varieties, and its sublattice of all group varieties. We conclude, therefore, that to completely describe this lattice, we must first describe completely the lattice of all structurally trivial semigroup varieties, since the other two are well known lattices.     

Semigroups whose Proper Right Congruences Form a Tree

Let S be a semigroup whose set of proper right congruences form a tree. The main theorem is a characterization of those semigroups having this property. In this characterization we draw on the results of Schein and Tamura for commutative semigroups and Kozhukhov for left chain semigroups and Hitzel for nilpotent semigroups. The interested reader should also see the work of Nagy on -semigroups.     

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.