Browsing by Author "Bakare, Gatta N."

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    ON THE SEMIGROUP RING OF THE RHOTRIX BICYCLIC SEMIGROUP
    (Published by Federal University of Technology, Minna., 2026-04-14) Ahmed, Bayo Musa; Bakare, Gatta N.; Usamot, Idayat F.
    In this paper, we study the algebraic structure of the semigroup ring associated with the rhotrix bicyclic semigroup . Let be a ring and the rhotrix bicyclic semigroup defined on ordered pairs of sequences indexed by a fixed index set. We construct the semigroup ring and investigate some of its structural properties. In particular, we establish that is generally non-commutative and determine a family of idempotent elements arising from idempotents of . The ordering of these idempotents is described componentwise, and this ordering induces a natural hierarchy within the semigroup ring. Furthermore, several classes of subrings of are identified, including subrings generated by subsemigroups, idempotents, and corner subrings determined by idempotent elements. These results reveal how the structural properties of the rhotrix bicyclic semigroup influence the internal structure of its semigroup ring and provide a foundation for further study of ideals and related algebraic properties.
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    Subsemigroup Structure of the Rhotrix Bicyclic Semigroup
    (Published by Library and Publication Committee, University of Ilorin, Nigeria., 2026-04-14) Ahmed, Bayo Musa; Bakare, Gatta N.; Suleiman, Y. M.
    This paper investigates the subsemigroup structure of the rhotrix bicyclic semigroup obtained as a coordinatewise extension of the classical bicyclic semigroup. Order-defined subsemigroups determined by the componenetwise relations 􀝌􀝅􀵒􀝍􀝅,􀝌􀝅􀵑􀝍􀝅 and 􀝌􀝅=􀝍􀝅 are introduced and analyzed. It is shown that the order-decreasing subsemigroup forms a subsemigroup but is neither regular nor inverse, with its regular elements coinciding precisely with the diagonal elements. The order-increasing subsemigroup is established to be anti-isomorphic to the order-decreasing case, while the diagonal subsemigroup is proved to be commutative idempotent semigroup and is isomorphic to a semilattice under componentwise maximum. These results provide structural classification of order-defined subsemigroups of the rhotrix bicyclic semigroup. Keywords:Rhotrix bicyclic semigroup; order-defined subsemigroup; regularity; idempotent structure;