#### Title

#### Date of Award:

1963

#### Document Type:

Thesis

#### Degree Name:

Master of Science (MS)

#### Department:

Mathematics and Statistics

#### Advisor/Chair:

Charles H. Cunkle

#### Abstract

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 (x_{1}, x_{2}, ..., x_{n}) is associative if x_{i} ⋅ (x_{j} ⋅ x_{k}) = (x_{i} ⋅ x_{j}) ⋅ x_{k} 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 (x_{1}, x_{2}, ..., x_{n}) is associative if x_{i} ⋅ (x_{j} ⋅ x_{k}) /= (x_{i} ⋅ x_{j}) ⋅ x_{k} 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 x_{i} ⋅ (x_{j} ⋅ x_{k}) /= (x_{i} ⋅ x_{j}) ⋅ x_{k } 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 x_{i} ⋅ (x_{j} ⋅ x_{k}) /= (x_{i} ⋅ x_{j}) ⋅ x_{k } 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.

#### Recommended Citation

Rogers, Dick R., "Anti-Associative Systems" (1963). *All Graduate Theses and Dissertations*. 6800.

https://digitalcommons.usu.edu/etd/6800

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