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 .
Partial algebraic systems sit between algebraic systems and relational systems; they are generalizations of algebraic systems, but special cases of relational systems.
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.
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1.
Every algebraic system is automatically a partial algebraic system.
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A division ring is a prototypical example of a partial algebra that is not an algebra. It has type . It is not an algebra because the unary operation (multiplicative inverse) is only partial, not defined for .
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Let be the set of all non-negative integers. Let “” be the ordinary subtraction. Then is a partial algebra.
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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.
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A small category can also be thought of as a partial algebra of type , where the two (total) unary operators are the source and target operations.
Remark. Like algebraic systems, one can define subalgebras, direct products, homomorphisms, as well as congruences in partial algebras.
References
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
Title | partial algebraic system |
Canonical name | PartialAlgebraicSystem |
Date of creation | 2013-03-22 18:42:10 |
Last modified on | 2013-03-22 18:42:10 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 28 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 03E99 |
Classification | msc 08A55 |
Classification | msc 08A62 |
Synonym | partial operator |
Synonym | partial algebra |
Related topic | RelationalSystem |
Defines | partial operation |
Defines | partial groupoid |