Showing papers on "Cancellative semigroup published in 1992"

The Adjoint of a Semigroup of Linear Operators

Abstract: The adjoint semigroup.- The ?(X,X?)-topology.- Interpolation, extrapolation and duality.- Perturbation theory.- Dichotomy theorems.- Adjoint semigroups and the RNP.- Tensor products.- The adjoint of a positive semigroup.

Semigroup algebras of the full matrix semigroup over a finite field

Abstract: Let M denote the multiplicative semigroup of all n-by-n matrices over a finite field F and K a commutative ring with an identity element in which the characteristic of F is a unit. It is proved here that the semigroup algebra K[M] is the direct sum of n + 1 algebras, namely, of one full matrix algebra over each of the group algebras K[GL(r, F)] with r = 0, 1, ..., n . The degree of the relevant matrix algebra over K[GL(r, F)] is the number of r-dimensional subspaces in an n-dimensional vector space over F. For K a field of characteristic different from that of F, this result was announced by Faddeev in 1976. He only published an incomplete sketch of his proof, which relied on details from the representation theory of finite general linear groups. The present proof is self-contained.

Semigroups of left quotients―the uniqueness problem

Abstract: Let S be a subsemigroup of a semigroup Q. Then Q is a semigroup of left quotients of S if every element of Q can be written as a*b, where a lies in a group J^-class of Q and a* is the inverse of a in this group; in addition, we insist that every element of S satisfying a weak cancellation condition named square-cancellable lie in a subgroup of Q. J. B. Fountain and M. Petrich gave an example of a semigroup having two non-isomorphic semigroups of left quotients. More positive results are available if we restrict the classes of semigroups from which the semigroups of left quotients may come. For example, a semigroup has at most one bisimple inverse wsemigroup of left quotients. The crux of the matter is the restrictions to a semigroup S of Green's relations 3? and if in a semigroup of quotients of S. With this in mind we give necessary and sufficient conditions for two semigroups of left quotients of S to be isomorphic under an isomorphism fixing S pointwise. The above result is then used to show that if R is a subring of rings Q, and Q2 and the multiplicative subsemigroups of Q, and Q2 are semigroups of left quotients of the multiplicative semigroup of R, then Q, and Q2 are isomorphic rings.

The minimal degree of a finite inverse semigroup

Abstract: 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

The adjoint of a positive semigroup

Abstract: We study the properties of the adjoint of a positive semigroup T (t) of operators on a Banach lattice E. The main results are: (i) If x∗ ⊥ E , then lim supt↓0 ‖T ∗(t)x∗−x∗‖ > 2‖x∗‖; (ii) If x∗ ⊥ E and either E∗ has order continuous norm or E has a quasi-interior point, then T ∗(t)x∗ ⊥ x∗ for almost all t; (iii) If E∗ has order continuous norm, then E is a projection band; (iv) If T ∗(t) is a lattice semigroup, then the disjoint complement of E is T ∗(t)-invariant.

On multiplicative bases in commutative semigroups

Abstract: We generalize some older results on multiplicative bases of integers to a certain class of commutative semigroups. In particular, we examine the structure of union bases of integers.

On existence of a dual semigroup

Abstract: We establish the existence of a sub-Markov semigroup in a Borel state space, which is in duality to a given sub-Markov semigroup with respect to a given excessive measure. The only assumption is that the initial semigroup is normal and separates points.

The regular radical of semigroup rings of commutative semigroups

Abstract: A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ( R [ S ]) for each associative ring R and commutative semigroup S .