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.     

Characterization of Simplicity and Cancellativity in βS

We determine precisely when the Stone-Cech compactification βS of a discrete semigroup S is simple and when it is left cancellative or right cancellative. As a consequence we see that βS is cancellative only when it is trivially so. That is, βS is cancellative if and only if S is a finite group.     

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.     

The ring of quotients of R(S)

A ring of quotients of the semigroup ring R(S) is discussed where R has a σ-set Σ and S has a σ-set Δ. In particular, we study the cases where (1) R is an integral domain and S is a commutative cancellative semigroup, (2) R is a commutative ring and S is a semilattice and (3) R is a commutative ring and S is a Rees matrix semigroup over a semigroup. Communicated by G. Lallement     

Semigroups of right quotients of a semigroup which is a semilattice of groups

Let S be a semigroup with zero which is a semilattice of groups. In [6], McMorris showed that the semigroup of quotients Q=Q(S) corresponding to the filter of “dense” right ideals of the semigroup S is also a semilattice of groups. He accomplished this by noting that Q is a regular semigroup in which all idempotents are central, an equivalent formulation of a semilattice of groups. In this paper we develop the semigroup of quotients Q corresponding to an arbitrary right quotient filter on S (as defined herein) and note the above result in this more general setting by explicitly constructing a semigroup which is isomorphic to Q. We also see that the underlying semilattice for Q in this case is isomorphic to a semigroup of quotients of the original semilattice for the semigroup S.     

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.     

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band E_S of idempotents of S such that F∋e, E_S∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of F_S. Dedicated to Professor L. M. Gluskin on his 60th birthday     

Free completely <Emphasis Type="Italic">J</Emphasis><Superscript>(<Emphasis Type="Italic">ℓ</Emphasis>)</Superscript>-simple Semigroups

A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

Free Completely J~(e)-simple Semigroups

A semigroup is called completely J~((e))-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid.It is proved that completely J~((e))-simple semigroups form a quasivarr ity.Moreover,the construction of free completely J~((e))-simple semigroups is given.It is found that a free completely J~((e))-simple semigroup is just a free completely J~*-simple semigroup and also a full subsemigroup of some completely simple semigroups.     

Semigroups which contain magnifying elements are factorizable

A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma).

In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.     

*-orderable semigroups

Fix a *-orderable field k. We introduce the class of *-orderable semigroups as those semigroups with involution S for which the semigroup algebra kS endowed with the canonical involution admits a *-ordering. It is shown that this class is a quasivariety that is locally and residually closed. A cancellative nilpotent semigroup with involution is proved to be *-orderable if and only if it has unique extraction of roots. In general this equivalence fails, although every *-orderable semigroup has unique extraction of roots.     

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.     

On the embedding of topological domains into quotient fields

Necessary and sufficient conditions to embed a Hausdorff left Ore domain A in a topological field Q(A) of quotients are given in terms of conditions on an invariant uniformity topologizing the multiplicative semigroup A* of A. The categorical aspects of such an embedding are discussed, and a new proof of S. Warner's theorem on the embedding of a left Ore domain as an open subring of a topological field of quotients is given.This work comprises part of the author's doctoral dissertation submitted to SUNY at Buffalo. The author wishes to thank Professor Dov Tamari for suggesting this problem.     

The New Structure Theorem of Right-e Wlpp Semigroups

Wlpp semigroups are generalizations of lpp semigroups and regular semi-groups. In this paper, we consider some kinds of wlpp semigroups, namely right-e wlpp semigroups. It is proved that such a semigroup S , if and only if S is the strong semilattice of L-right cancellative planks;also if and only if S is a spined product of a right-e wlpp semigroup and a left normal band.     

具有中心幂等元的L-弱正则半群

本文定义了具有中心幂等元的(L)-弱正则半群,研究了这类半群的代数结构.利用半群上的右同余(L)+和左同余R+,证明了半群S是一个具有中心幂等元的(L)-弱正则半群,当且仅当S是H-左可消幺半群的强半格.这推广了Clifford半群的相应结果.     

Profinite Groups Admitting Just Infinite Quotients

 A profinite group is said to be just infinite if each of its proper quotients is finite. We address the question which profinite groups admit just infinite quotients. It is proved that any profinite group whose order (as a supernatural number) is divisible only by finitely many primes admits just infinite quotients. It is shown that if a profinite group G possesses the property in question then so does every open subgroup and every finite extension of G. Received 20 July 2001     

Some examples of semigroup algebras of finite representation type

The semigroup algebras over a field K of the semigroups T_n of all permutations of a set of n elements are considered. It is proved: if n≤3 and (n!)^-1∈ K then the algebra KT_n has a finite representation type. Also the finiteness of the representation type of the semigroup algebra KS is established, where S is the sub-semigroup of T_n (n is arbitrary) such that S=J_n∪G where J_n={x∈T_n|rank x=1}, while G is a doubly transitive subgroup of the symmetric group S_n, the order of G being invertible in K. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 229–238, 1987.     

On the Embedding of Ordered Semigroups into Ordered Group

It was shown in [7] that any right reversible, cancellative ordered semigroup can be embedded into an ordered group and as a consequence, it was shown that a commutative ordered semigroup can be embedded into an ordered group if and only if it is cancellative. In this paper we introduce the concept of L-maher and R-maher semigroups and use a technique similar to that used in [7] to show that any left reversible cancellative ordered L or R-maher semigroup can be embedded into an ordered group.     

Completions for partially ordered semigroups

A standard completion γ assigns a closure system to each partially ordered set in such a way that the point closures are precisely the (order-theoretical) principal ideals. If S is a partially ordered semigroup such that all left and all right translations are γ-continuous (i.e., Y∈γS implies {x∈S:y·x∈Y}∈γS and {x∈S:x·y∈Y}∈γS for all y∈S), then S is called a γ-semigroup. If S is a γ-semigroup, then the completion γS is a complete residuated semigroup, and the canonical principal ideal embedding of S in γS is a semigroup homomorphism. We investigate the universal properties of γ-semigroup completions and find that under rather weak conditions on γ, the category of complete residuated semigroups is a reflective subcategory of the category of γ-semigroups. Our results apply, for example, to the Dedekind-MacNeille completion by cuts, but also to certain join-completions associated with so-called “subset systems”. Related facts are derived for conditional completions. A first draft of this paper by the second author, containing parts of Section 2, was received on August 9, 1985.     

t-Sets and some algebraic properties in β S and in ℓ ∈ fty(S) *

For a large class of infinite discrete semigroups, we prove that right cancellative points in β S can have arbitrary norms or sizes. More precisely, if for x∈β S, we let ||x||= min{|A| : x ∈