On the Presentation and Cayley Graph of the Bruck–Reilly Idealization Semigroup

Transferring constructions between different algebraic structures often reveals deep connections and enables the application of techniques from one theory to another. The idealization of the module over a ring, introduced by Nagata in 1962, has been a powerful tool in commutative algebra for decades. Recently, Wazzan and Ozalan developed a semigroup analogue of this idealization structure, denoted by SB1⊛B, using Bruck–Reilly extensions, opening new avenues for exploring the interaction between semigroup theory and ring theory. In this paper, we establish a presentation–theoretic and graph–theoretic framework for the new semigroup structure described above. More specifically, we prove that the finite presentation is transferred to the idealization SB1⊛B from the base monoid, construct a strongly connected Cayley graph with infinite diameter, and establish that the word problem for SB1⊛B is solvable when it is solvable for the base monoid. In line with the classical results in ring theory, some results in this paper show that certain fundamental properties are preserved under a special idealization of semigroups.