Volume 60, Issue 4

The concept of coloring is studied for graphs derived from lattices with 0. It is shown that, if such a graph is derived from an atomic or distributive lattice, then the chromatic number equals the clique number. If this number is finite, then in the case of a distributive lattice, it is determined by the number of minimal prime ideals in the lattice. An estimate for the number of edges in such a graph of a finite lattice is given.

CBA logic was introduced as a non-associative generalization of the Łukasiewicz many-valued propositional logic. Its algebraic semantic is just the variety of commutative basic algebras. Petr Hájek introduced vt-operators as models for the “very true” connective on fuzzy logics. The aim of the paper is to show possibilities of using vt-operators on commutative basic algebras, especially we show that CBA logic endowed with very true connective is still fuzzy.

For an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

In the paper, we extend Binet’s first formula for the logarithm of the gamma function and investigate some properties, including inequalities, star-shaped and sub-additive properties and the complete monotonicity, of the extended remainder of Binet’s first formula for the logarithm of the gamma function and related functions.

Making use of the familiar convolution structure of analytic functions, we introduce a general class of multivalently analytic functions and derive various useful properties and characteristics of this function class by using the techniques of differential subordination. Several other results are presented exhibiting relevant connections to some of the results obtained in earlier works.

The singular boundary value problems of third-order differential equations $$ \begin{array}{*{20}c} { - u'''(t) = h(t)f(t,u(t)), t \in (0,1),} \\ {u(0) = u'(0) = 0, u'(1) = \alpha u'(\eta )} \\ \end{array} $$ are considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where h(t) is allowed to be singular at both t = 0 and t = 1, and f is not necessary to be nonnegative. The existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory.

A lower triangular infinite matrix is called a triangle if there are no zeros on the principal diagonal. The main result of this paper gives a minimal set of sufficient conditions for a double triangle T to be a bounded operator on ; i.e., T ∈ B () for the sequence space defined below. As special summability methods T we consider weighted mean and double Cesàro, (C, 1, 1), methods. As a corollary we obtain necessary and sufficient conditions for a double triangle T to be a bounded operator on the space of double sequences of bounded variation.

In [HOLÁ, Ľ.—HOLÝ, D.: Pointwise convergence of quasicontinuous mappings and Baire spaces, Rocky Mountain J. Math.] a complete answer is given, for a Baire space X, to the question of when the pointwise limit of a sequence of real-valued quasicontinuous functions defined on X is quasicontinuous. In [HOLÁ, Ľ.—HOLÝ, D.: Minimal USCO maps, densely continuous forms and upper semicontinuous functions, Rocky Mountain J. Math. 39 (2009), 545–562], a characterization of minimal USCO maps by quasicontinuous and subcontinuous selections is proved. Continuing these results, we study closed and compact subsets of the space of quasicontinuous functions and minimal USCO maps equipped with the topology of pointwise convergence. We also study conditions under which the closure of the graph of a set-valued mapping which is the pointwise limit of a net of set-valued mappings, is a minimal USCO map.

We introduce a lower semicontinuous analog, L −(X), of the well-studied space of upper semicontinuous set-valued maps with nonempty compact interval images. Because the elements of L −(X) contain continuous selections, the space C(X) of real-valued continuous functions on X can be used to establish properties of L −(X), such as the two interrelated main theorems. The first of these theorems, the Extension Theorem, is proved in this Part I. The Extension Theorem says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). The second main theorem, the Factorization Theorem, is proved in Part II. The Factorization Theorem says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.

This is a continuation of “Spaces of lower semicontinuous setvalued maps I”. Together, these two parts contain two interrelated main theorems. In the previous part I, the Extension Theorem is proved, which says that for binormal spaces X and Y, every bimonotone homeomorphism between C(X) and C(Y) can be extended to an ordered homeomorphism between L −(X) and L −(Y). In this part II, the Factorization Theorem is proved, which says that for binormal spaces X and Y, every ordered homeomorphism between L −(X) and L −(Y) can be characterized by a unique factorization.