Volume 58, Issue 6

A semi-ℓg-cone is an algebra (C;*,:,·) of type (2, 2, 2) satisfying the equations (a*a)*b = b = b: (a: a); a*(b: c) = (a*b): c; a: (b*a) = (b: a)*b and (ab) *c = b* (a * c). An ℓ-group cone is a semi-ℓg-cone and a bounded semi-ℓg-cone is term equivalent to a pseudo MV-algebra. Also, a subset A of a semi-_g-cone C is an ideal of C if and only if it is a deductive system of its reduct (C;*,:).

We prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.

It is shown that every difference poset can be converted into a total algebra in a manner similar to that which is used for difference lattices. As a tool applied here we have commutative directoids and posets with sectional antitone involutions.

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same sets of symbols. This survey paper summarizes the theory of latin bitrades, detailing their applications to critical sets, random latin squares and existence constructions for latin squares.

We use the concept of generalized MV-algebra (GMV-algebra, in short) in the sense of Galatos and Tsinakis; the main tool in their investigation was a truncation construction. The relations between radical classes of GMV-algebras and radical classes of lattice ordered groups are investigated in the present paper. Further, we apply the truncation construction for dealing with weak retract mappings of GMV-algebras.

In this paper we investigate sequential convergences on a cyclically ordered group G which are compatible with the group operation and with the relation of cyclic order; we do not assume the validity of the Urysohn’s axiom. The system convG of convergences under consideration is partially ordered by means of the set-theoretical inclusion. We prove that convG is a Brouwerian lattice.

The aim of this paper is to prove some random fixed point theorems for asymptotically nonexpansive random operator defined on an unbounded closed and starshaped subset of a Banach space.

A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology.