THE CONCEPT OF RHOTRIX BICYCLIC SEMIGROUP

RHOTRIX BICYCLIC SEMIGROUP

Abstract

 We introduce and study a rhotrix analogue of the classical bicyclic semigroup. Let $B(\mathbb{N}_0)$ denote the set of ordered pairs of rhotrices with entries in the non-negative integers. A binary operation is defined componentwise via a scalar minimum adjustment rule extending the bicyclic multiplication on $\mathbb{N}_0^2$. Associativity is first established at the scalar level and then lifted to rhotrices, yielding a well-defined semigroup. We show that $B(\mathbb{N}_0)$ forms a non-commutative monoid with identity, prove that every element is regular, and characterize the idempotents as precisely the element of the form $ \big(P, P\big)$. In addition, we identify a natural closed subsemigroup determined by a coordinatewise order condition. These results provide a rigorous semigroup-theoretic foundation for rhotrix based algebraic structures and extend classical bicyclic constructions to the rhotrix setting.