Matrix rank 1 semigroup identities

A finite basis of identities is constructed for the semigroup of all rank 1 n × n matri­ces over the field. It is worthy to notice that every semigroup of all rank r, r > l,n×n matrices over a finite field has no finite basis of identities. Let G be an arbitrary vari­ety of groups with a finite basis of identities. A finite basis of identities is constructed for the variety generated by all completely 0-simple semigroups over G-groups.     

The diophantine equation (a n-1)(b n-1)=x 2

Let a and b given unequal positive integers; it is desired to determine the positive integer solutions n and x of the equation of the title. Some special cases have recently been considered, and here some general results and conjectures are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Rank properties of the semigroup of endomorphisms over Brandt semigroup

We investigate small rank, lower rank, intermediate rank, upper rank, and large rank of the semigroup of endomorphisms over Brandt semigroup.     

Every semigroup is isomorphic to a transitive semigroup of binary relations

Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).

     

Every numerical semigroup is one half of a symmetric numerical semigroup

Let be a numerical semigroup. Then there exists a symmetric numerical semigroup such that .

     

A note on the power semigroup of a completely simple semigroup

We prove the pseudovariety generated by power semigroups of completely simple semigroups is the semidirect product of the pseudovariety of block groups with the pseudovariety of right zero semigroups, and hence is decidable. This answers a question of Almeida from over 15 years ago. The author was supported in part by NSERC.     

正规Ehresmann型wrpp半群

众所周知,Clifford半群是正则半群类中的一类重要半群,本文定义正规 Ehresmann型wrpp半群,它是Clifford半群在wrpp半群类中的推广,给出了此类半群的若干刻划.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).