In this paper, we present the $ \Psi $-intersection graph $ \Psi_{\mathcal{S}}(\mathcal{B}) $ associated with an $ S $-act over topological semigroups. We investigate how the internal structure of $ S $-acts, particularly the presence and distribution of simple subacts, determines fundamental graph-theoretic properties of $ \Psi_{\mathcal{S}}(\mathcal{B}) $. Characterizations of connectivity, bipartiteness, girth, and diameter are established in terms of decomposition properties and chain conditions on subacts. We further analyze the clique structure of $ \Psi_{\mathcal{S}}(\mathcal{B}) $, showing that every clique contains at most one simple subact and describing maximal cliques through families of non-simple subacts intersecting via simple components. Domination-type parameters are also studied by introducing indecomposable controlling sets and deriving bounds for the control number of $ \Psi_{\mathcal{S}}(\mathcal{B}) $. Finally, by using monophonic convexity and the $ m $-topology on graphs, we examine induced path structures and their interaction with algebraic paths in $ S $-acts, revealing connections between geodesic behavior and semigroup actions. These results provide a unified algebraic, topological, and graph-theoretic framework and open new combinatorial and metric approaches to the study of topological path semigroups.
Citation: Husniyah Alzubaidi. On the P-intersection graphs of acts over semigroups: applications to connectivity and monophonic convexity with clique structure[J]. AIMS Mathematics, 2026, 11(7): 20619-20644. doi: 10.3934/math.2026839
In this paper, we present the $ \Psi $-intersection graph $ \Psi_{\mathcal{S}}(\mathcal{B}) $ associated with an $ S $-act over topological semigroups. We investigate how the internal structure of $ S $-acts, particularly the presence and distribution of simple subacts, determines fundamental graph-theoretic properties of $ \Psi_{\mathcal{S}}(\mathcal{B}) $. Characterizations of connectivity, bipartiteness, girth, and diameter are established in terms of decomposition properties and chain conditions on subacts. We further analyze the clique structure of $ \Psi_{\mathcal{S}}(\mathcal{B}) $, showing that every clique contains at most one simple subact and describing maximal cliques through families of non-simple subacts intersecting via simple components. Domination-type parameters are also studied by introducing indecomposable controlling sets and deriving bounds for the control number of $ \Psi_{\mathcal{S}}(\mathcal{B}) $. Finally, by using monophonic convexity and the $ m $-topology on graphs, we examine induced path structures and their interaction with algebraic paths in $ S $-acts, revealing connections between geodesic behavior and semigroup actions. These results provide a unified algebraic, topological, and graph-theoretic framework and open new combinatorial and metric approaches to the study of topological path semigroups.
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Husniyah Alzubaidi. On the P-intersection graphs of acts over semigroups: applications to connectivity and monophonic convexity with clique structure[J]. AIMS Mathematics, 2026, 11(7): 20619-20644. doi: 10.3934/math.2026839
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