Evaluation of Some Algebraic Structures in Balanced Incomplete Block Design

Abstract

Balanced incomplete block design is an incomplete block design in which any two varieties appear together an equal number of times. In algebra, the existence of block design is closely related to balanced incomplete block design. To ascertain the claim, this research aim to employ some algebraic structures to examine whether or not balanced incomplete block design is related to the above statement. The methods adopted are finite group, ring and field algebra. The result shows that balanced incomplete block design (BIBD) cannot form a finite group under multiplication binary operation, but it is for additive case. It is also revealed that balanced incomplete block design is not a field algebra in both binary operations no matter the size of the design, but it is a ring in all cases. In conclusion, BIBD of the form (X,B) is a semigroup, commutative group, semiring, commutative ring and subfield in both binary operations. Several theorems with proofs have been established in harmony with the algebraic structure mentioned above.