Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Representations of inverse semigroups by one-to-one partial transformations of a set

We show that each representation ϕ, say, of an inverse semigroup S, by means of transformations of a set X, determines a representation ϕ^* by means of partial one-to-one transformations of X, in such a fashion that sϕ ↦ sϕ^*, for s ∈ S, is an isomorphism of Sϕ upon Sϕ^*. An immediate corollary is the classical faithful representation of an inverse semigroup as a semigroup of partial one-to-one transformations.     

Semigroups with the Congruence Extension Property

ρT on a semigroup of T of S extends to the semigroup S, if there exists a congruence _ρ on s such that _{ρ|T= ρT}. A semigroup S has the congruence extension property, CEP, if each congruence on each semigroup extends to S. In this paper we characterize the semigroups with CEP by a set of conditions on their structure (by this we answer a problem put forward in [1]). In particular, every such semigroup is a semilattice of nil extensions of rectangular groups.     

On Weighted Hardy Spaces on Complex Semigroups

H^p (S,α) on a complex open Ol'shanskii semigroup S = G Exp (iW), where 1 ≤p≤∞ and α is an absolute value on the involutive semigroup X. For 1 < p < ∞ we prove the existence of an isometric boundary value map H ^p (S,α) → L ^p (G) generalizing the corresponding result of Ol'shanskii for p = 2 and α = 1. In the second part we use the fine structure of the space H ² (S,1) to prove the existence of a bounded holomorphic function on S whose absolute value has a unique maximum in the boudary point 1Β G and therefore complete the proof of the approximation property of the Poisson kernel and the uniqueness of G as a Shilov boundary of S whenever W does not contain affine line.     

Baer extensions of rings and stone extensions of semigroups

Let R be a commutative semigroup [resp. ring] with identity and zero, but without nilpotent elements. We say that R is a Stone semigroup [Baer ring], if for each annihilator ideal P⊂R there are idempotents e₁ ε P and e₂ ε Ann(P) such that x→(e₁x, e₂x):R→P×Ann(P) is an isomorphism. We show that for a given R there exists a Stone semigroup [Baer ring] S containing R that is minimal with respect to this property. In the ring case, S is uniquely determined if one requires that there be a natural bijection between the sets of annihilator ideals of R and S. This is close to results of J. Kist [5]. Like Kist, we use elementary sheaf-theoretical methods (see [2], [3], [6]). Proofs are not very detailed. An address delivered at the Symposium on Semigroups and the Multiplicative Structure of Rings, University of Puerto Rico, Mayaguez, Puerto Rico, March 9–13, 1970.     

The Similarity Problem for Bounded Analytic Semigroups on Hilbert Space

t )_t≥0 on a Hilbert space H, we establish conditions under which (T_t)_t≥0 is similar to a contraction semigroup, i.e., there exists an isomorphism S Ε B (H) such that (S^-1 T_t S)_t≥0 is a contraction semigroup. In the case when the generator -A of (T_t)_t≥0 is one-to-one, we obtain that (T_t)_t≥0 is similar to a contraction semigroup if and only if A admits bounded imaginary powers. This characterizes one-to-one operators of type strictly less than π/2 on H which belong to BIP (H).     

Embedding topological semigroups in topological groups

In [6] Rothman investigated the problem of embedding a topological semigroup in a topological group. He defined a concept calledProperty F and showed that Property F is a necessary and sufficient condition for embedding a commutative, cancellative topological semigroup in its group of quotients as an open subset. This paper announces a generalization of Rothman’s result by definingProperty E and stating that a completely regular topological semigroup S can be embedded in a topological group by a topological isomorphism if and only if S can be embedded (algebraically) in a group and S has Property E. Property E is defined by first constructing a free topological semigroup (Theorem 1.1). This construction resembles the one in [4] for a free topological group. Full details, examples, and other embedding results will appear elsewhere. Some of the results in this paper were contained in the author’s doctoral dissertation written at Rutgers University under Professor Louis F. McAuley.     

Medial permutable semigroups of the first kind

We say that a semigroup S is a permutable semigroup if the congruences of S commute with each other, that is, α○β=β○α is satisfied for all congruences α and β of S. A semigroup is called a medial semigroup if it satisfies the identity axyb=ayxb. The medial permutable semigroups were examined in Proc. Coll. Math. Soc. János Bolyai, vol. 39, pp. 21–39 (1981), where the medial semigroups of the first, the second and the third kind were characterized, respectively. In Atta Accad. Sci. Torino, I-Cl. Sci. Fis. Mat. Nat. 117, 355–368 (1983) a construction was given for medial permutable semigroups of the second [the third] kind. In the present paper we give a construction for medial permutable semigroups of the first kind. We prove that they can be obtained from non-archimedean commutative permutable semigroups (which were characterized in Semigroup Forum 10, 55–66, 1975). Research supported by the Hungarian NFSR grant No T042481 and No T043034.     

Homomorphisms between Stone-Cech Homomorphisms into Stone-Cech Remainders of Countable Groups

We study properties of continuous homomorphisms from β S into T^* and from S^* into T^*, where S denotes a countably infinite semigroup and T denotes a countably infinite group. We show that they have striking algebraic properties if they do not arise as continuous extensions of homomorphisms from S to T.     

On Lattice Isomorphisms of Inverse Semigroups, II

A lattice isomorphism between inverse semigroups S and T is an isomorphism between their lattices of inverse subsemigroups. When S is combinatorial, it has long been known that a bijection is induced between S and T. Various authors have introduced successively weaker "archimedean" hypotheses under which this bijection is necessarily an isomorphism, naturally inducing the original lattice isomorphism. Since lattice-isomorphic groups need not have the same cardinality, extending these techniques to the non-combinatorial case requires some means of tying the subgroups to the rest of the semigroup. Ershova showed that if S has no nontrivial isolated subgroups (subgroups that form an entire D-class) then again a bijection exists between S and T. Recently, this technique has been successfully exploited, by Goberstein for fundamental inverse semigroups and by the author for completely semisimple inverse semigroups, under two different finiteness hypotheses. In this paper, we derive further properties of Ershova's bijection(s) and formulate a "quasi-connected" hypothesis that enables us to derive both Goberstein's and the author's earlier results as corollaries.     

Non-group Ranks in Finite Full Transformation Semigroups

T _n be the full transformation semigroup on a finite set. Both rank and idempotent rank of the semigroup K(n,r) = {α∈T _n : | im α | ≤r, 2 ≤ r ≤ n - 1. In this paper we prove that the non-group rank, defined as the cardinality of a minimal generating set of non-group elements, of K(n,r) is S(n,r) , the Stirling number of the second kind.     

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra ℓ¹(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then ℓ¹(S) is approximately amenable if and only if G is amenable. Moreover ℓ¹(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every M_a(S)-measurable subset B of S and s ∈ S the set sB is M_a(S)-measurable, it is proved that if the measure algebra M_a(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.     

On Products of Hypercyclic Semigroups

If (S(t)) is a hypercyclic (discrete or continuous) semigroup of linear operators, it is known that (S(t) ⊗ S(t)) is hypercyclic, if and only if (S(t)) satisfies the so-called hypercyclicity criterion (HCC). We give a strengthened version of the hypercyclicity criterion, which we call recurrent hypercyclicity criterion (RHCC). It is a necessary and sufficient condition on a semigroup (S(t)), such that the product with any semigroup (T(t)) satisfying HCC is again hypercyclic. The RHCC is inherited by products (obviously) and by subsemigroups. Any chaotic semigroup satisfies the RHCC.     

The chain of right simple semigroups in ordered semigroups

The relation ≤ is defined on the set of right ideals of an ordered semigroup. The main result of this paper is as follows: an ordered semigroup S is a chain of right simple ordered semigroups if and only if ≤ is an order relation. Bibliography: 3 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 83–88.     

Nilpotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of a finite semigroup S is the size of a minimal generating set of S, and the nilpotent rank of S is the size of a minimal generating set of S consisting of nilpotents in S. A partition of a q-element subset of the set X_n = {1,2,..., n} is said to be of type τ if the sizes of its classes form the partition τ of the positive integer q ≤ n. A non-trivial partition τ of q consists of k < q elements. For a non-trivial partition τ of q < n, the semigroup S(τ), generated by all the transformations with kernels of type τ, is nilpotent-generated. We prove that if τ is a non-trivial partition of q < n, then the rank and the nilpotent rank of S(τ) are both equal to the number of partitions X_n of type τ.     

Relative universality and universality obtained by adding constants

A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety \mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in \mathbbW{\mathbb{W}} is algebraically universal. And \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in \mathbbV{\mathbb{V}} gives rise to a class a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture.     

O-simple strictly regular semigroups

In the previous paper [6], it has been proved that a semigroup S is strictly regular if and only if S is isomorphic to a quasi-direct product EX Λ of a band E and an inverse semigroup Λ. The main purpose of this paper is to present the following results and some relevant matters: (1) A quasi-direct product EX Λ of a band E and an inverse semigroup Λ is simple [bisimple] if and only if Λ is simple [bisimple], and (2) in case where EX Λ has a zero element, EX Λ is O-simple [O-bisimple] if and only if Λ is O-simple [O-bisimple]. Any notation and terminology should be referred to [1], [5] and [6], unless otherwise stated.     

Guarded and Banded Semigroups

The variety of guarded semigroups consists of all (S,·, ˉ) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g_⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g_⋆ ≅ L is obtained.     

Transformational antiatomic characteristic of ordered sets

To any ordered set with a universally maximal element, a semigroup of its transformations with some natural properties that defines the ordered set up to an isomorphism is assigned. The system of such transformation semigroups is proved to be the minimal element in the set of all defining systems of transformation semigroups with respect to the following ordering: one system precedes another if for each ordered set from the class in question, the semigroup of its transformation belonging to the first system is contained in the semigroup of its transformation from the second system. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 112–119, July, 1999.