partial algebraic systemPartialAlgebraicSystem2009-01-05 21:01:102009-01-12 16:00:07Definitionalgebraic systempartial operationpartial groupoid\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}Let $\lambda$ be a cardinal. A partial function $f: A^{\lambda} \to A$ is called a \emph{partial operation} on $A$. $\lambda$ is called the arity of $f$. When $\lambda$ is finite, $f$ is said to be \emph{finitary}. Otherwise, it is \emph{infinitary}. A nullary partial operation is an element of $A$ and is called a constant.
\textbf{Definition}. A \emph{partial algebraic system} (or \emph{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 \emph{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 \emph{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.
\begin{enumerate}
\item
Every algebraic system is automatically a partial algebraic system.
\item
A division ring $(D,\lbrace +\mbox{, }\cdot\mbox{, }-\mbox{, }^{-1}\mbox{, }0\mbox{, }1\rbrace)$ 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 $0$.
\item
Let $A$ be the set of all non-negative integers. Let ``$-$'' be the ordinary subtraction. Then $(A,\lbrace -\rbrace)$ is a partial algebra.
\item
A \emph{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 \PMlinkname{groupoids (category theoretic)}{GroupoidCategoryTheoretic}, and Brandt groupoids, all of which are partial.
\item
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.
\end{enumerate}
\textbf{Remark}. Like algebraic systems, one can define subalgebras, direct products, homomorphisms, as well as congruences in partial algebras.
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\bibitem{gg} G. Gr\"{a}tzer: {\em Universal Algebra}, 2nd Edition, Springer, New York (1978).
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