manysorted structure
Let $L$ be a manysorted language and $S$ the set of sorts. A manysorted structure $M$ for $L$, or simply an $L$structure^{} consists of the following:

1.
for each sort $s\in S$, a nonempty set ${A}_{s}$,

2.
for each function symbol $f$ of sort type $({s}_{1},\mathrm{\dots},{s}_{n})$:

3.
for each relation symbol $r$ of sort type $({s}_{1},\mathrm{\dots},{s}_{n})$, a relation^{} (or subset)
$${r}_{M}\subseteq {A}_{{s}_{1}}\times \mathrm{\cdots}\times {A}_{{s}_{n}}.$$
A manysorted algebra is a manysorted structure without any relations.
Remark. A manysorted structure is a special case of a more general concept^{} called a manysorted interpretation, which consists all of items 13 above, as well as the following:

4.
an element ${x}_{M}\in {A}_{s}$ for each variable $x$ of sort $s$.
Examples.

1.
A left module over a ring can be thought of as a twosorted algebra^{} (say, with sorts $\{{s}_{1},{s}_{2}\}$), for there are

–
there are two nonempty sets $M$ (corresponding to sort ${s}_{1}$) and $R$ (corresponding to sort ${s}_{2}$), where

–
$M$ has the structure of an abelian group^{} (equipped with three operations^{}: $0,,+$, corresponding to function symbols of sort types $({s}_{1}),({s}_{1},{s}_{1})$, and $({s}_{1},{s}_{1},{s}_{1})$)

–
$R$ has the structure of a ring (equipped with at least four operations: $0,,+,\times $, corresponding to function symbols of sort types $({s}_{2}),({s}_{2},{s}_{2})$ and $({s}_{2},{s}_{2},{s}_{2})$ for $+$ and $\times $, and possibly a fifth operation $1$ of sort type $({s}_{2})$)

–
a function $\cdot :R\times M\to M$, which corresponds to a function symbol of sort type $({s}_{2},{s}_{1},{s}_{1})$. Clearly, $\cdot $ is the scalar multiplication on the module $M$.
For a right module over a ring, one merely replaces the sort type of the last function symbol by the sort type $({s}_{1},{s}_{2},{s}_{1})$.

–

2.
A deterministic^{} semiautomaton $A=(S,\mathrm{\Sigma},\delta )$ is a twosorted algebra, where

–
$S$ and $\mathrm{\Sigma}$ are nonempty sets, corresponding to sorts, say, ${s}_{1}$ and ${s}_{2}$,

–
$\delta :S\times \mathrm{\Sigma}\to S$ is a function corresponding to a function symbol of sort type $({s}_{1},{s}_{2},{s}_{1})$.

–

3.
A deterministic automaton $B=(S,\mathrm{\Sigma},\delta ,\sigma ,F)$ is a twosorted structure, where

–
$(S,\mathrm{\Sigma},\delta )$ is a semiautomaton discussed earlier,

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$\sigma $ is a constant corresponding to a nullary function symbol of sort type $({s}_{1})$,

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$F$ is a unary relation corresponding to a relation symbol of sort type $({s}_{1})$.
Because $F$ is a relation, $B$ is not an algebra.

–

4.
A complete sequential machine $M=(S,\mathrm{\Sigma},\mathrm{\Delta},\delta ,\lambda )$ is a threesorted algebra, where

–
$(S,\mathrm{\Sigma},\delta )$ is a semiautomaton discussed earlier,

–
$\mathrm{\Delta}$ is a nonempty sets, corresponding to sort, say, ${s}_{3}$,

–
$\lambda :S\times \mathrm{\Sigma}\to \mathrm{\Delta}$ is a function corresponding to a function symbol of sort type $({s}_{1},{s}_{2},{s}_{3})$.

–
References
 1 J. D. Monk, Mathematical Logic, Springer, New York (1976).
Title  manysorted structure 
Canonical name  ManysortedStructure 
Date of creation  20130322 17:45:17 
Last modified on  20130322 17:45:17 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03B70 
Classification  msc 03B10 
Classification  msc 03C07 
Synonym  many sorted structure 
Synonym  many sorted algebra 
Defines  manysorted interpretation 
Defines  manysorted algebra 