partial algebraic system

Let λ be a cardinal. A partial functionMathworldPlanetmath f:AλA is called a partial operation on A. λ is called the arity of f. When λ is finite, f is said to be finitary. Otherwise, it is infinitary. A nullary partial operation is an element of A and is called a constant.

Definition. A partial algebraic system (or partial algebra for short) is defined as a pair (A,O), where A is a set, usually non-empty, and called the underlying set of the algebraMathworldPlanetmathPlanetmathPlanetmath, and O is a set of finitary partial operations on A. The partial algebra (A,O) is sometimes denoted by 𝑨.

Partial algebraic systems sit between algebraic systems and relational systemsMathworldPlanetmath; they are generalizationsPlanetmathPlanetmath 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 𝑨 of type τ, we typically mean that 𝑨 is proper, meaning that the partial operation f𝑨 is non-empty for every function symbol fτ, and if f is a constant symbol, f𝑨A.

Below is a short list of partial algebras.

  1. 1.

    Every algebraic system is automatically a partial algebraic system.

  2. 2.

    A division ring (D,{+--101}) is a prototypical example of a partial algebra that is not an algebra. It has type 2,2,1,1,0,0. It is not an algebra because the unary operation -1 (multiplicative inverse) is only partial, not defined for 0.

  3. 3.

    Let A be the set of all non-negative integers. Let “-” be the ordinary subtraction. Then (A,{-}) is a partial algebra.

  4. 4.

    A partial groupoid is a partial algebra of type 2. In other words, it is a set with a partial binary operationMathworldPlanetmath (called the productMathworldPlanetmathPlanetmathPlanetmath) on it. For example, a small category may be viewed as a partial algebra. The product ab is only defined when the source of a matches with the target of b. Special types of small categories are groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (categoryMathworldPlanetmath theoretic) (, and Brandt groupoids, all of which are partial.

  5. 5.

    A small category can also be thought of as a partial algebra of type 2,1,1, where the two (total) unary operators are the source and target operationsMathworldPlanetmath.

Remark. Like algebraic systems, one can define subalgebrasMathworldPlanetmathPlanetmath, direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, as well as congruencesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath in partial algebras.


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