partial algebraic system
Let be a cardinal. A partial function is called a partial operation on . is called the arity of . When is finite, is said to be finitary. Otherwise, it is infinitary. A nullary partial operation is an element of and is called a constant.
Definition. A partial algebraic system (or partial algebra for short) is defined as a pair , where is a set, usually non-empty, and called the underlying set of the algebra, and is a set of finitary partial operations on . The partial algebra is sometimes denoted by .
The type of a partial algebra is defined exactly the same way as that of an algebra. When we speak of a partial algebra of type , we typically mean that is proper, meaning that the partial operation is non-empty for every function symbol , and if is a constant symbol, .
Below is a short list of partial algebras.
Every algebraic system is automatically a partial algebraic system.
Let be the set of all non-negative integers. Let “” be the ordinary subtraction. Then is a partial algebra.
A partial groupoid is a partial algebra of type . In other words, it is a set with a partial binary operation (called the product) on it. For example, a small category may be viewed as a partial algebra. The product is only defined when the source of matches with the target of . Special types of small categories are groupoids (category theoretic) (http://planetmath.org/GroupoidCategoryTheoretic), and Brandt groupoids, all of which are partial.
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
|Title||partial algebraic system|
|Date of creation||2013-03-22 18:42:10|
|Last modified on||2013-03-22 18:42:10|
|Last modified by||CWoo (3771)|