partial algebraic system
Let $\lambda $ be a cardinal. A partial function^{} $f:{A}^{\lambda}\to A$ is called a partial operation on $A$. $\lambda $ is called the arity of $f$. When $\lambda $ 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 nonempty, and called the underlying set of the algebra^{}, and $O$ is a set of finitary partial operations on $A$. The partial algebra $(A,O)$ is sometimes denoted by $\bm{A}$.
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 $\bm{A}$ of type $\tau $, we typically mean that $\bm{A}$ is proper, meaning that the partial operation ${f}_{\bm{A}}$ is nonempty for every function symbol $f\in \tau $, and if $f$ is a constant symbol, ${f}_{\bm{A}}\in A$.
Below is a short list of partial algebras.

1.
Every algebraic system is automatically a partial algebraic system.

2.
A division ring $(D,\{+\text{,}\cdot \text{,}{\text{,}}^{1}\text{,}0\text{,}1\})$ is a prototypical example of a partial algebra that is not an algebra. It has type $\u27e82,2,1,1,0,0\u27e9$. It is not an algebra because the unary operation ${}^{1}$ (multiplicative inverse) is only partial, not defined for $0$.

3.
Let $A$ be the set of all nonnegative integers. Let “$$” be the ordinary subtraction. Then $(A,\{\})$ is a partial algebra.

4.
A partial groupoid is a partial algebra of type $\u27e82\u27e9$. 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 $ab$ is only defined when the source of $a$ matches with the target of $b$. Special types of small categories are groupoids^{} (category^{} theoretic) (http://planetmath.org/GroupoidCategoryTheoretic), and Brandt groupoids, all of which are partial.

5.
A small category can also be thought of as a partial algebra of type $\u27e82,1,1\u27e9$, 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  20130322 18:42:10 
Last modified on  20130322 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 