Multialgebra

Builds on Set Power set Arity Cartesian product

Definition

Let $\Sigma$ be a signature, a set of function symbols each with an arity $n \ge 0$ . For a set $A$ , let $\mathcal{P}^*(A)$ be the set of nonempty subsets of $A$ , a part of the power set. A multialgebra of signature $\Sigma$ is a set $A$ together with a map $f^A \colon A^n \to \mathcal{P}^*(A)$ for each symbol $f \in \Sigma$ of arity $n$ . Here $A^n$ is the Cartesian power of $A$ . Each map $f^A$ is a multioperation on $A$ .

Remarks

An ordinary algebra is the same as a multialgebra in which every value $f^A(a_1, \dots, a_n)$ has exactly one element. Some authors allow the empty set as a value; then partial algebras are also multialgebras.

Example: $A = \R$ with the binary multioperation $x \circ y = [\min(x, y), \max(x, y)]$ . Each value is a closed interval that contains $x$ , so it is nonempty.

Non-example: $A = \set{0, 1, 2}$ with $x \circ y = \set{z \in A : z > x + y}$ . For $x = y = 1$ the value is $\set{z \in A : z > 2} = \emptyset$ , so $\circ$ is not a multioperation in the sense above.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Arity concept
Cartesian product concept
Element definition
Power set concept
Set definition
Multialgebra definition

Tag	`Y4AQFH6N3Y`
URI	`m2://Y4AQFH6N3Y`
Kind	definition
Writing	W0 stub
MSC	08A99, 20N20
Defines	Multialgebra
Uses	Set, Power set, Arity, Cartesian product

Multialgebra

Definition

Remarks

Reading path

Further reading

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