Boris M. Schein

Relation algebras and function semigroups

The concept of relation algebra unifies many familiar notions from algebra (especially those of systems having “natural” models as groups, Boolean algebras etc.). The fundamental theorem on relation algebras asserts the existence of simple conditions which characterize any given class of relation algebras.The idea of relation algebra is very useful for the study of function and transformation semigroups, which is the central part of the theory of semigroups. It provides a general outlook, permits one to formulate many natural problems, and ensures that these problems possess non-trivial solutions. A number of examples illustrate this point.

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or mirror inner. In particular, we answer a question of B. I. Plotkin.

A combinatorial property of finite full transformation semigroups

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

Endomorphisms of finite full transformation semigroups

We describe all endomorphisms of finite full transformation semigroups and count their number. A full transformation semigroup TX on a set X is the set X of all transformations (i.e., self-maps) X → X of X with composition of transformations as multiplication. This is an important object in semigroup theory, combinatorics, many-valued logic, etc. Various properties of TX are known. In particular, Schreier [4] proved in 1936 that automorphisms of TX are inner: for every automorphism α there exists a uniquely determined element g ∈ GX ⊂ TX of the symmetric group GX on X such that α(t) = gtg−1 for all t ∈ TX . Here the juxtaposition gt stands for the composition g ◦ t and the composition acts from the right to the left: g◦t(x) = g(t(x)) for every x ∈ X . Thus the automorphism group of TX is naturally isomorphic to GX . Surprisingly, no one has considered endomorphisms of TX . Our paper seems to be the first attempt at filling that gap. We consider the finite case only, that is, X is a finite set of cardinality n for n ≥ 0. We introduce a few notations and terms. Endomorphisms that are not automorphisms are called proper. The kernel congruence ker(ε) of an endomorphism ε is defined by (s, t) ∈ ker(ε) ⇔ ε(s) = ε(t) for any s, t ∈ TX . ∆A is the identity relation on a set A. If ε′ = ε|GX is the restriction of ε to GX , then ker(ε′) also stands for the corresponding normal subgroup of GX . The second projection (also called the range) of t ∈ TX is the set pr2 t = t(X). In particular, pr2(st) ⊂ pr2 s. The rank of t is the cardinality | pr2(t)| of pr2(t). We can assume that X = {1, 2, . . . , n} and write Tn instead of TX . Analogously, Gn stands for GX . We consider Gn as a subgroup of Gn+1 consisting of all permutations that fix the point n+1. Also, AX denotes the alternating group on X . For n = 3 or n ≥ 5, An is the only nontrivial normal subgroup of Gn, while G4 contains another nontrivial normal subgroup K, Klein’s four-group. For every x ∈ X , cx denotes the constant transformation in TX that maps all elements of X onto x. For example, c4 = ( 1 2 3 4 4 4 4 4 ) in T4. Our main results are the following Theorem and Corollary. Their proof is followed by a Proposition that is another corollary to our main theorem. Received by the editors February 12, 1997. 1991 Mathematics Subject Classification. Primary 20M20; Secondary 03G25, 05A15. c ©1998 American Mathematical Society

THE SEMIGROUP OF ONE-TO-ONE TRANSFORMATIONS WITH FINITE DEFECTS

Let if be the semigroup of all total one-to-one transformations of an infinite set X. For an / e V let the deject of/, def/, be the cardinality of X R(J), where R(f) =f(X) is the range of /. Then Sf is a disjoint union of the symmetric group ^x on X, the semigroup 5 of all transformations in y with finite non-zero defects and the semigroup S of all transformations in 5 with infinite defects, such that 5 U 5 and S are ideals of Sf. The properties of *3X a d S have been investigated by a number of authors (for the latter it was done via Baer-Levi semigroups, see [2], [3], [5], [6], [7], [8], [9], [10] and note that S decomposes into a union of Baer-Levi semigroups). Our aim here is to study the semigroup S. It is not difficult to see that S is left cancellative (we compose functions /, g in 5 as/g(x) =/(g(*)), for x e X) and idempotent-free. All automorphisms of 5 are inner [4], that is of the form f^hfh~\f€S,he <SX. In the present paper, we are concerned with congruences, Greens relations and ideals of 5. A large variety of distinct types of congruences on S is present and the main results are the content of Theorems 4, 5 and 6. In the concluding remark we state some unsolved problems and conjectures on congruences on 5. For/, g e S, let D(J, g) = {* :f(x)3=g(x)}. The next lemma is easily verified.

Automorphisms of the Endomorphism Semigroup of a Free Inverse Semigroup

We prove that automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between the endomorphism semigroups of free inverse semigroups.

Bands with isomorphic endomorphism semigroups

Let End S denote the endomorphism semigroup of a semigroup S. If S and T are bands (i.e., idempotent semigroups), then every isomorphism of End S onto End T is induced by a uniquely determined bijection of S onto T. This bijection either preserves both the natural order relation and D-quasi-order relation on S or reverses both these relations. If S is a normal band, then the bijection induces isomorphisms of all rectangular components of S onto those of T, or it induces anti-isomorphisms of the rectangular components. If both S and T are normal bands, then S and T are isomorphic, or anti-isomorphic, or S ≅ Y × C and T ≅ Y∗ × C, where Y is a chain considered as a lower semilattice, Y∗ is a dual chain, and C is a rectangular band, or S ≅ Y × C, T ≅ Y∗ × Copp, where Y, Y∗, and C are as above and Copp is antiisomorphic to C.

The minimal degree of a finite inverse semigroup

The minimal degree of an inverse semigroup S is the minimal cardinality of a set A such that S is isomorphic to an inverse semigroup of one-to-one partial transformations of A. The main result is a formula that expresses the minimal degree of a finite inverse semigroup S in terms of certain subgroups and the ordered structure of S. In fact, a representation of S by one-to-one partial transformations of the smallest possible set A is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries

Interconnections between the structure theory of set addition and rewritability in groups

An approach to groups and semigroups stemming from the structure theory of set addition turns out to have much in common with the so-called permutation or rewritable properties. We explain these connections and show how these properties take their place in a wider class of interesting and naturally arising problems. As an example, we characterize some classes of groups and

Semigroups of tolerance relations

Abstract We give an abstract algebraic characterization of semigroups of tolerance relations and semigroups of symmetric binary relations.