ON WHETHER A t- STEINER QUINTUPLE SYSTEM OF BALANCED INCOMPLETE BLOCK DESIGN IS A GROUP, RING OR FIELD ALGEBRA - Isaac, Anthony Asuquo; Akra, Ukeme Paulinus; Akpan, Stephen S. (FUDMA Journal of Sciences, 2025)
This paper seeks to establish if the t- Steiner quintuple system of balanced incomplete block design (BIBD) is a group, ring or field algebra. A 2- (11, 5, 2) BIBD was constructed with its blocks, incidence matrix and Cayley table shown and the axioms of the algebraic structures of group, ring and field defined.  The t- Steiner quintuple design, represented as, GE was tested with the axioms of the algebraic structures. The results showed that a t – Steiner quintuple balanced incomplete block design obeys all the axioms of a group under the additive binary operation but breaks down under the multiplicative binary operation. Results further showed that while the t-Steiner quintuple design satisfied all the axioms of a Ring algebra, hence is a ring, a semi-group, a commutative semi-ring and a commutative ring, but failed to obey the axioms of a Field algebra, hence, it is not a field algebra.
Keywords: Algebraic Structure; Balanced Designs; Binary Operation; Steiner Quintuple Designs