Anti-Associative Systems
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...
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| Format: | Others |
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DigitalCommons@USU
1963
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| Online Access: | https://digitalcommons.usu.edu/etd/6800 https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=7851&context=etd |
| Summary: | 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. |
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