Partial algebra

Remarks

When $\operatorname{dom}(f^A) = A^n$ for every $f$, the partial algebra is an ordinary (total) algebra.

Example: the natural numbers $\N$ with subtraction, where $m - n$ is defined exactly when $m \ge n$. The domain is $\set{(m, n) \in \N^2 : m \ge n}$, and on it $m - n \in \N$, so subtraction is a partial operation on $\N$. It is not total, since $(0, 1)$ is not in its domain.

Non-example: subtraction on all of $\N^2$ with values in $\Z$ is not a partial operation on $\N$, since its value $0 - 1 = -1$ does not lie in the carrier $\N$.

A field $K$ with the partial operation $x \mapsto x^{-1}$ defined on $K \setminus \set{0}$ is a partial algebra for the signature $\set{+, \cdot, -, ^{-1}, 0, 1}$. Homomorphisms, subalgebras and congruences admit several inequivalent versions for partial algebras, for example weak and strong homomorphisms, so the version in use must be stated.