Many-sorted algebra

Builds on Set Function Cartesian product

Definition

Let $S$ be a set, whose elements are called sorts. An $S$ -sorted signature $\Sigma$ is a set of operation symbols, each with a type $s_1 \cdots s_n \to s$ , where $n \ge 0$ and $s_1, \dots, s_n, s \in S$ . A many-sorted $\Sigma$ -algebra $A$ consists of:

a set $A_s$ for each sort $s \in S$ , the carrier of sort $s$ ;
for each symbol $f$ of type $s_1 \cdots s_n \to s$ , a function $f^A \colon A_{s_1} \times \cdots \times A_{s_n} \to A_s$ on the Cartesian product.

A homomorphism $h \colon A \to B$ of $\Sigma$ -algebras is a family of maps $h_s \colon A_s \to B_s$ with $h_s(f^A(a_1, \dots, a_n)) = f^B(h_{s_1}(a_1), \dots, h_{s_n}(a_n))$ for every symbol $f$ of type $s_1 \cdots s_n \to s$ and all $a_i \in A_{s_i}$ .

Remarks

Example: let $S = \set{\mathrm{scalar}, \mathrm{vector}}$ , with symbols $+ \colon \mathrm{vector}\,\mathrm{vector} \to \mathrm{vector}$ , $\cdot \colon \mathrm{scalar}\,\mathrm{vector} \to \mathrm{vector}$ , and the field operations on the sort $\mathrm{scalar}$ . A vector space $V$ over a field $K$ gives an algebra with $A_{\mathrm{scalar}} = K$ and $A_{\mathrm{vector}} = V$ ; each operation sends arguments of the right sorts to the right sort.

Non-example: the same data with a map $K \times V \to K$ in place of scalar multiplication fails clause 2, since the value of $\cdot$ must lie in the carrier of sort $\mathrm{vector}$ .

With one sort, a many-sorted algebra is an ordinary algebra. Birkhoff and Lipson called these structures heterogeneous algebras. Some authors require each carrier $A_s$ to be nonempty.

Reading path

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Cartesian product concept
Element definition
Function concept
Set definition
Many-sorted algebra definition

Tag	`ZETPP2VJFJ`
URI	`m2://ZETPP2VJFJ`
Kind	definition
Writing	W0 stub
MSC	08A68, 68Q65
Defines	Many-sorted algebra
Uses	Set, Function, Cartesian product
Changed	2 November 2026

Many-sorted algebra

Definition

Remarks

Reading path

Further reading

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