Representations of distributive lattice-ordered semigroups with binary relations

LetR(, , ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(, , ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, , , ) where (A, , ) is a distributive lattice, (A, ) is a semigroup and is additive w.r.t. . We prove thatDLOS is the variety generated byR(, , ¦), and moreover, if (A, , , ) DLOS then it is representable whenever we disregard one of its operations.Presented by Boris M. Schein.Research supported by Hungarian National Foundation for Scientific Research grant No. 1810.     

Periodic lattice-ordered pregroups are distributive

It is proved that any lattice-ordered pregroup that satisfies an identity of the form x ^ll···l = x ^rr···r (for the same number of ^l,r -operations on each side) has a lattice reduct that is distributive. It follows that every such ?-pregroup is embedded in an ?-pregroup of residuated and dually residuated maps on a chain.     

On lattice-ordered commutative semigroups

Decompositions of elements into intersections of primal elements and into intersections of p-components are studied in certain lattice-ordered commutative semigroups, by making use of the new development in commutative ideal theory without finiteness conditions, due to Fuchs-Heinzer-Olberding [7]. Several results concerning ideals can be phrased as theorems in abstract ideal theory.The intersections we consider are in general not irredundant, and the associated prime elements are not unique. However, one can establish a canonical intersection that is often irredundant with uniquely determined associated primes.     

Completely decomposable lattice-ordered semigroups

Siberian Mathematical Journal -     

Undecidability of representability for lattice-ordered semigroups and ordered complemented semigroups

We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with respect to a universal relation, this result can be extended to infinite representations of ordered complemented semigroups.     

On the distributive radical of an Archimedean lattice-ordered group

Let G be an Archimedean ℓ-group. We denote by G ^d and R _D (G) the divisible hull of G and the distributive radical of G, respectively. In the present note we prove the relation (R _D (G)) ^d = R _D (G ^d ). As an application, we show that if G is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.     

Feedback boundary control problems for linear semigroups

We investigate the abstract Cauchy problem and apply the obtained generation results to feedback boundary control problems.     

Algorithmic problems in varieties of semigroups

Translated from Algebra i Logika, Vol. 27, No. 4, pp. 440–463, July–August, 1988.     

Some key problems on finite semigroups

The paper starts from a historical perspective of the theory of finite semigroups, namely of its most developed part, the theory of pseudovarieties and its connections with the theories of finite automata and rational languages. It goes on to review some of the most important developments and to present some of the key problems that have guided the theory for the past 35 years. Several open problems that may in turn play a significant role in years to come are also discussed. May 31, 2001     

Distribution semigroups and abstract Cauchy problems

We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator in a Banach space the following assertions are equivalent: (a) generates a distribution semigroup; (b) the convolution operator has a fundamental solution in where denotes the domain of supplied with the graph norm and denotes the inclusion ; (c) generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

     

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065–3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491–509, 2010) and many others.     

Unsolvable algorithmic problems for semigroups,groups, and rings

A survey of the subject outlined in the heading (with many proof s sketched) is given. A special focus is on the original proofs of the unsolvability theorems of Markov, Post, and Novikov for word problems in semigroups and groups. A method of Shirshov is described, which has led to proof of the main unsolvability theorems for Lie algebras.Translated from Itogi Nauki I Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 25, pp. 3–66, 1987.The Russian names in this survey are given according to standard transliteration. Some Russian authors, however, are also known in the Western literature in a different spelling. The corresponding equivalents are given below, in alphabetical order: