The semi-direct product of arbitrary two monoids and a presentation for this product have received considerable attention, see for instance [12], [14] and [15]. In [15], Wang defined a trivialiser set of the Squier complex associated with this presentation. In this paper, as a main result, we discuss necessary and sufficient conditions for the standard presentation of the semi-direct product of...

In generalized semi-infinite programming the feasible set is known to be not closed in general. In this paper, under natural and generic assumptions, the closure of the feasible set is described in explicit terms.

We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...

This paper deals with a new class of nonlinear set valued implicit variational inclusion problems involving (A, η)-monotone mappings in 2-uniformly smooth Banach spaces. Semi-inner product structure has been used to study the (A, η)-monotonicity. Using the generalized resolvent operator technique and the semi-inner product structure, the approximation solvability of the proposed problem is inve...