in this paper, the notion of maximal m-open set is introduced and itsproperties are investigated. some results about existence of maximal m-open setsare given. moreover, the relations between maximal m-open sets in an m-spaceand maximal open sets in the corresponding generated topology are considered.our results are supported by examples and counterexamples.

Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

Very little is known about the structure of intersection two $k$-generated monoids words, even for $k=3$. Here we investigate case $k$-maximal monoids, that is, whose basis cardinality $k$ cannot be non-trivially decomposed into at most words. We characterize in $3$-maximal monoids.

In this paper, the notion of maximal m-open set is introduced and itsproperties are investigated. Some results about existence of maximal m-open setsare given. Moreover, the relations between maximal m-open sets in an m-spaceand maximal open sets in the corresponding generated topology are considered.Our results are supported by examples and counterexamples.

Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of 54 -tough maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995), 301–305]. In addition, we present two generalizations of a similar result of Tkáč who considered 1-tough maximal planar graphs [Discrete Math. 154 (1996), 321–328]; we rem...