Anti-Associative Systems

• Main Author: Rogers, Dick R.
• Format: Others
• Published: DigitalCommons@USU 1963
• Subjects: anti-associative systems; elements; semi-associative systems; Discrete Mathematics and Combinatorics; Mathematics
• Online Access: https://digitalcommons.usu.edu/etd/6800; https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=7851&context=etd

A set of elements with a binary operation is called a system, or, more explicitly, a mathematical system. The following discussion will involve systems with only one operation. This operation will be denoted by "⋅" and will sometimes be referred to as a product. A system, S, of n elements (x1, x2, ..., xn) is associative if xi ⋅ (xj ⋅ xk) = (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n. In a modern algebra class the following problem was proposed. What is the least number of elements a system can have and be non-associative? A system, S, of n elements (x1, x2, ..., xn) is associative if xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for some i, j, k ≤ n. It is obvious that a system of one element must be associative. Any binary operation could have but one result. A nonassociative system of two elements (a, b) can be constructed by letting a ⋅ a = b⋅a = b. , a⋅(a⋅a) = a⋅b and (a⋅a)⋅a = b⋅a = b. If a⋅b = a, then a⋅(a⋅a) /= (a⋅a)⋅a Thus the system is nonassociative. As is often the case this question leads to others. Are there systems of n elements such that xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n? If such systems exist, what are their charcateristics? Such questions as these led to the development of this paper. A system, S, of n elements such that xi ⋅ (xj ⋅ xk) /= (xi ⋅ xj) ⋅ xk for all i, j, k ≤ n is called an anti-associative system. The purpose of this paper is to establish the existence of antiassociative systems of n elements and to find characteristics of these systems in as much detail as possible. Propositions will first be considered that apply to anti-associative systems in general. Then anti-associative systems of two, three, and four elements will be obtained. The general results that each of these special cases lead to will be developed. A special type of anti-associative system will be considered. These special anti-associative systems suggest a broader field. For a set of elements a group of classes of systems is defined. The operation may associative, anti-associative, or neither. Many questions are let unanswered as to the characteristics of anti-associative systems, but this paper opens new avenues to attack a broader problem.