Nilpotents in semigroups of partial one-to-one order-preserving mappings

Communicated by R. B. McFadden     

Representations of semigroups of partial isometries

We present a new proof that the irreducible representationsof the von Neumann algebra generated by a strongly continuoussemigroup of partial isometries of index 1 are unique up toequivalence, as well as a proof that when such an algebra isa factor, its representations are completely reducible. As anapplication, we show that the irreducible representations ofa strongly continuous semigroup of isometries {U(), 0} suchthat are equivalent.     

Completely regular semigroups as semigroups of partial transformations

Communicated by G. Lallement     

Representations of a set with two partial order relations

Suppose that I is a finite set with two partial order relations > and relative to each of which its width is at most two. We designate as a representation of I a pair of matrices A,B over any field having an equal number of columns and similarly partitioned into vertical bands, indexed by the elements of I. We shall regard two matrix pairs with the same structure as equivalent if one can pass from one pair to the other by means of a sequence of transformations of the following types: 1) nonsingular linear transformations of the rows of the matrices A,B; 2) nonsingular linear transformations of the columns of any vertical band simultaneously in both matrices; 3) addition of linear combinations of columns of the matrix A from the band numbered i to columns of the band numbered j>i without changing B; 4) addition of linear combinations of columns of the matrix B from the band numbered i to columns of the band numbered ji, without changing the matrix A. A classification of all matrix pairs to within this equivalence is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 69–75, 1983.     

Bases of identities for semigroups of bounded rank transformations of a set

We complete the series of results by M. V. Sapir, M. V. Volkov and the author solving the Finite Basis Problem for semigroups of rank ≤ k transformations of a set, namely based on these results we prove that the semigroup T _k (X) of rank ≤ k transformations of a set X has no finite basis of identities if and only if k is a natural number and either k = 2 and |X| ∈ «3, 4» or k ≥ 3. A new method for constructing finite non-finitely based semigroups is developed. We prove that the semigroup of rank ≤ 2 transformations of a 4-element set has no finite basis of identities but that the problem of checking its identities is tractable (polynomial).     

Actions of inverse semigroups arising from partial actions of groups

In this work we present a definition of crossed product for actions of inverse semigroups on C^∗-algebras, without resorting to covariant representations as done by Sieben in related work. We also show the existence of an isomorphism between the crossed product by a partial action of a group G and the crossed product by a related action of S(G), an inverse semigroup associated to G introduced by the first named author.     

On inverse semigroups the closure of whose set of idempotents is a clifford semigroup

Communicated by Boris Schein     

Representations of involutive semigroups

Involutive [topological] semigroups are semigroups endowed with an involutive antiautomorphism. A representation of such a semigroup means an involution preserving [weakly continuous] morphism into the algebra of bounded operators on a Hilbert space. We develop a representation theory of involutive [topological] semigroups based on positive definite functions on them. We do not generally assume the existence of an identity element. This makes the proofs of some results harder, but most results hold in this general setting. The author thanks the referee for many constructive suggestions to improve the exposition of the paper.     

Defining relations of semigroups of all directed transformations of an ordered finite set

The semigroup of all transformations X of a finite (partially) ordered set , such that X for all , is considered. All possible generating sets of a are elucidated. Only one of those sets is irreducible. A system of defining relations is found for that generating set.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 657–662, June, 1968.     

关于反射等价关系的变换半群的注记

设TX是非空集合X上全变换半群,E是X上非平凡的等价关系,则T?(X)是TX的子半群.在赋予半群T?(X)自然偏序关系的条件下,本文刻画了它的相容元.