Factor-powers of finite symmetric groups

To a transformation semigroup (S, M) we assign a new semigroupFP(S) called the factor-power of the semigroup (S, M). Then we apply this construction to the symmetric groupS _n. Some combinatorial properties of the semigroupFP(S _n) are studied; in particular, we investigate its relationship with the semigroup of 2-stochastic matrices of ordern and the structure of its idempotents. The idempotents are used in characterizingFP(S _n) as an extremal subsemigroup of the semigroupB _n of all binary relations of ann-element set and also in the proof of the fact thatFP(S _n) contains almost all elements ofB _n.Translated fromMatematicheskie Zametki, Vol. 58, No. 2, pp. 176–188, August, 1995.This work was partially supported by the Foundation for Basic Research of the Ukrainian State Committee on Science and Technology.     

On permutation properties in groups and semigroups

A semigroupS satisfiesPPn, thepermutation property of degree n (n≥2) if every product ofn elements inS remains invariant under some nontrivial permutation of its factors. It is shown that a semigroup satisfiesPP3 if and only if it contains at most one nontrivial commutator. Further a regular semigroup is a semilattice ofPP3 right or left groups, and a subdirect product ofPP3 semigroups of a simple type. A negative answer to a question posed by Restivo and Reutenauer is provided by a suitablePP3 group.     

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.     

On some problems of Petrich concerning Bruck and Reilly semigroups

For an inverse semigroupS with its idempotents dually well-ordered, we prove thatS is isomorphic to the semigroup of all one-to-one partial right translations ofS. Also, we prove for a Bruck semigroupS=B(T, α) thatS isE-unitary if and only ifT isE-unitary and α is an idempotent pure homomorphism. Moreover, we characterize allE-unitary covers ofB(T, α), whereT is a finite chain of groups.     

Su una condizione di finitezza per semigruppi finitamente generati

We prove that a finitely generated semigroupS is finite if and only if there exists an integern such that in each sequence ofn elements ofS there exist two different non empty factors with the same value inS. We prove this result using only elementary facts concerning the canonical form of an element of a finitely generated semigroup.     

On the heredity of Hun and Hungarian property

Denote byD(S) the convolution semigroup of compact-regular probability measures on a topological semigroupS. Hincin's classical decomposition theorems are extended to finite point processes on a completely regular topological space and to the convolution semigroupsD(D(G)), D(D(D(G))),... whereG is a locally compact Hausdorff group. The paper applies the Hun-Hungarian semigroup theory approach of Ruzsa and Székely; the proofs also follow this abstract setting.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.     

The universal group of homogeneous quotients

This paper constructs from the homogeneous quotients of an arbitrary semigroupS a universal group (G(S), γ) onS. If S is left reversible and cancellative, thenG(S) coincides with the embedding group of quotients of S due to Ore. If S is an inverse semigroup, G(S) coincides with the maximum group homomorphic image of S due to Munn. In these cases, γ coincides with the embedding and canonical homomorphism respectively ofS intoG(S). In general (G(S), γ) is equivalent to the universal group on S due to N. Bouleau. A universal group constructed from the set of Lambek ratios had earlier been exhibited by A.H. Clifford and G.B. Preston for cancellative semigroups satisfying the condition Z of Malcev. No previous construction has, however, emerged as a direct generalisation of both the work of Ore and Munn as does the present. Elementary properties of homogeneous quotients are employed to illuminate Bouleau's counter-example on why certain Malcev conditions are insufficient to guarantee the embeddability of a semigroup in a group.     

Structure of a finite commutative inverse semigroup and a finite bundle for which the inverse monoid of local automorphisms is permutable

For a semigroup S, the set of all isomorphisms between the subsemigroups of the semigroup S with respect to composition is an inverse monoid denoted by PA(S) and called the monoid of local automorphisms of the semigroup S. The semigroup S is called permutable if, for any couple of congruences ρ and σ on S, we have ρ ∘ σ = σ ∘ ρ. We describe the structures of a finite commutative inverse semigroup and a finite bundle whose monoids of local automorphisms are permutable.     

Range sets and partition sets in connection with congruences and algebraic invariants

With a semigroup of transformationsS, we associate a class of equivalence relations onR(S) (calledclosed under inclusion relations), the set of ranges ofS. We define a new notion of connectedness for semigroups of transformations (calledrange-connectedness). For a range-connectedS, the closed under inclusion relations and the left-zero congruences ofS are dually isomorphic. The ideas above are dualized for the partition sets ofS. We associate withS an ordered pair which measures its range and partition connectedness. We generalize to an arbitrary semigroupT by considering faithful representations ofT by semigroups of transformations. In so doing, we are able to define an algebraic invariant for semigroups.     

Maximizing a congruence with respect to its partition of idempotents

For a congruence σ on a semigroupS a congruence μ(σ) onS, containing σ, is defined such that the semigroupS/σ is fundamental if and only if σ=μ(σ). The congruence μ(σ) is shown to possess maximality properties and for idempotent-surjective semigroups, μ(σ) is the maximum congruence with respect to the partition of the idempotents determined by σ. Thus μ is the maximum idempotent-separating congruence on any idempotent-surjective semigroup. It is shown that μ(μ(σ))=μ(σ). If ρ is another congruence onS, possibly with the same partition of the idempotents as σ, then it is of interest to know when ρ⊆σ (or ρ⊆μ(σ)) implies μ(ρ)⊆μ(σ) or even μ(ρ)=μ(σ). These implications are not true in general but if σ⊆ρ⊆μ(σ) then μ(ρ)⊆μ(σ). IfS is an idempotent-surjective semigroup and ρ and σ have the same partition of the idempotents then μ(ρ)=μ(σ).     

Remarks on maximal left ideals in semigroups

Some similar results to those for maximal (two-sided) ideals in a compact semigroupS are obtained for maximal left ideals inS, with one exception i.e. the intersection of all maximal left ideals inS may be empty. The maximal left ideals in the convolution semigroup of measures onS are also considered.     

Existence varieties with lattices of regular subsemigroups

A class of regular semigroups is called an existence variety, ore-variety, if it is closed under taking homomorphic images, regular subsemigroups, and direct products. For a regular semigroupS, the set of all regular subsemigroups ofS forms a partially ordered set under set inclusion. We determine for whiche-varietiesV the set of regular subsemigroups of members ofV forms a lattice. This includes the known result that the regular subsemigroups of an orthodox semigroup form a lattice.Presented by R. Freese.     

A Note on Semigroup Algebras of Permutable Semigroups

Let S be a semigroup and 𝔽 be a field. For an ideal J of the semigroup algebra 𝔽[S] of S over 𝔽, let ?_J denote the restriction (to S) of the congruence on 𝔽[S] defined by the ideal J. A semigroup S is called a permutable semigroup if α ○ β = β ○ α is satisfied for all congruences α and β of S. In this paper we show that if S is a semilattice or a rectangular band then φ_{{S; 𝔽}}: J → ?_J is a homomorphism of the semigroup (Con(𝔽[S]); ○ ) into the relation semigroup (?_S; ○ ) if and only if S is a permutable semigroup.     

The Circle Semigroup of a Ring

Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed. This revised version was published online in July 2006 with corrections to the Cover Date.     

TC semigroups and inflations

An algebraA satisfiesTC (the term condition) if for any and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS _E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS _Reg (the set of regular elements ofS) andS _Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.     

Retracts and monomorphism monoids in semigroup varieties

If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.     

Unary Semigroups with an Associate Subgroup

A subgroup H of a regular semigroup S is said to be an associate subgroup of S if for every s ∈ S, there is a unique associate of s in H. An idempotent z of S is said to be medial if czc = c, for every c product of idempotents of S. Blyth and Martins established a structure theorem for semigroups with an associate subgroup whose identity is a medial idempotent, in terms of an idempotent generated semigroup, a group and a single homomorphism. Here, we construct a system of axioms which characterize these semigroups in terms of a unary operation satisfying those axioms. As a generalization of this class of semigroups, we characterize regular semigroups S having a subgroup which is a transversal of a congruence on S.     

Some remarks on the structure of free automata

In this paper we define automata-linearly independence. An automatonM has a basis B iffM is free provided that we assume that the action ofS onX × S is (x,s)a = (x,sa) for alla, s ∈ S andx ∈X. If a semigroupS is PRID, every subautomaton of a freeS-automaton is free.     

Star-operations on semigroups

Let S be a grading monoid with quotient group q(S) , let F(S) be the set of fractional ideals of S . For A ∈ F(S) , define A _w = {x ∈ q(S) \mid J+x \subseteq A for some f.g. ideal J of S with J ^-1 =S} and A_ \overline w ={x ∈ q(S)\mid J+x \subseteq A for some ideal J of S with J ^-1 =S} . Then w and \overline w are star-operations on F(S) such that w ≤ \overline w . Using these star-operations, we give characterizations of Krull semigroups and pre-Krull semigroups. Also we show that for every maximal * -ideal P of S , if S _P is a valuation semigroup, then * -cancellation ideals are * -locally principal ideals, where * is a star-operation on S of finite character. Finally, we show that S is a pre-Krull semigroup (H-semigroup) if and only if the polynomial semigroup S[x] is a pre-Krull semigroup (H-semigroup). October 15, 1999