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 non-empty, 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 $𝑨$.

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 $𝑨$ of type $\tau$, we typically mean that $𝑨$ is proper, meaning that the partial operation $f_{𝑨}$ is non-empty for every function symbol $f\in \tau$, and if $f$ is a constant symbol, $f_{𝑨}\in A$.

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 $⟨2,2,1,1,0,0⟩$. It is not an algebra because the unary operation ${}^{-1}$ (multiplicative inverse) is only partial, not defined for $0$.

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

4. A partial groupoid is a partial algebra of type $⟨2⟩$. 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 $⟨2,1,1⟩$, 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.