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many-sorted structure
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(Definition)
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Let be a many-sorted language and the set of sorts. A many-sorted structure for , or simply an -structure consists of the following:
- for each sort
, a non-empty set ,
- for each function symbol
of sort type
:
- for each relation symbol
of sort type
, a relation (or subset)
Remark. A many-sorted structure is a special case of a more general concept called a many-sorted interpretation, which consists all of items 1-3 above, as well as the following:
- 4.
- an element
for each variable of sort .
Example. a left module over a ring can be thought of as a two-sorted structure (say, with sorts
), for there are
- there are two non-empty sets
(corresponding to sort ) and (corresponding to sort ), where
has the structure of an abelian group (equipped with three operations: , corresponding to function symbols of sort types
, and
)
has the structure of a ring (equipped with at least four operations:
, corresponding to function symbols of sort types
and
for and , and possibly a fifth operation of sort type )
- a function
, which corresponds to a function symbol of sort type
. Clearly, is the scalar multiplication on the module .
For a right module over a ring, one merely replaces the sort type of the last function symbol by the sort type
.
More to come...
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"many-sorted structure" is owned by CWoo.
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(view preamble)
| Other names: |
many sorted structure |
| Also defines: |
many-sorted interpretation |
This object's parent.
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Cross-references: scalar, operations, abelian group, structure, ring, left module, variable, subset, relation, relation symbol, constant symbol, function, sort type, function symbol, sorts, many-sorted language
This is version 3 of many-sorted structure, born on 2008-01-22, modified 2008-01-25.
Object id is 10206, canonical name is ManySortedStructure.
Accessed 408 times total.
Classification:
| AMS MSC: | 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic) | | | 03C07 (Mathematical logic and foundations :: Model theory :: Basic properties of first-order languages and structures) | | | 03B70 (Mathematical logic and foundations :: General logic :: Logic in computer science) |
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Pending Errata and Addenda
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