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# 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 $\boldsymbol{A}$.

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

Below is a short list of partial algebras.

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

2. A division ring $(D,\{+\mbox{, }\cdot\mbox{, }-\mbox{, }^{{-1}}\mbox{, }0\mbox{, }1\})$ is a prototypical example of a partial algebra that is not an algebra. It has type $\langle 2,2,1,1,0,0\rangle$. 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 $\langle 2\rangle$. 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), and Brandt groupoids, all of which are partial.5. A small category can also be thought of as a partial algebra of type $\langle 2,1,1\rangle$, 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).

## Mathematics Subject Classification

03E99*no label found*08A55

*no label found*08A62

*no label found*

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