# Dictionary Definition

substructure

### Noun

1 the basic structure or features of a system or
organization [syn: infrastructure]

2 lowest support of a structure; "it was built on
a base of solid rock"; "he stood at the foot of the tower" [syn:
foundation, base, fundament, foot, groundwork, understructure]

# User Contributed Dictionary

## English

### Noun

- The supporting part of a structure (either physical or organizational; the foundation).
- The earth or gravel that railway sleepers are embedded in.

### See also

# Extensive Definition

In universal
algebra, an (induced) substructure or (induced) subalgebra is a
structure whose domain is a subset of that of a bigger
structure, and whose functions and relations are the traces of the
functions and relations of the bigger structure. Some examples of
subalgebras are subgroups, submonoids, subrings, subfields, subalgebras of
algebras
over a field, or induced subgraphs.

In the presence of relations (i.e. for structures
such as ordered
groups or graphs,
whose signature
is not functional) it may make sense to relax the conditions on a
subalgebra so that the relations on a weak substructure (or weak
subalgebra) are at most those induced from the bigger structure.
Subgraphs
are an example where the distinction matters, and the term
"subgraph" does indeed
refer to weak substructures. Ordered
groups, on the other hand, have the special property that every
substructure of an ordered group which is itself an ordered group,
is an induced substructure.

In mathematical
logic, especially in model
theory, the term "submodel" is often used as a synonym for
substructure, in the same way that the term "model"
is used as a synonym for "structure". But often it has a slightly
more restrictive meaning described below.

## Definition

Given two
structures A and B of the same signature
σ, A is said to be a weak substructure of B, or a weak subalgebra
of B, if

- the domain of A is a subset of the domain of B,
- f A = f B | An for every n-ary function symbol f in σ, and
- R A \subseteq R B \cap An for every n-ary relation symbol R in σ.

A is said to be an (induced) substructure of B,
or an (induced) subalgebra of B, if A is a weak subalgebra of B
and, moreover,

- R A = R B \cap An for every n-ary relation symbol R in σ.

## Example

In the language consisting of the binary
functions + and ×, binary relation <, and
constants 0 and 1, the structure (Q, +, ×, <, 0,
1) is a substructure of (R, +, ×, <, 0, 1). More
generally, the substructures of an ordered
field (or just a field) are precisely its
subfields. Similarly, in the language (×, -1, 1) of
groups, the substructures of a group
are its subgroups. In
the language (×, 1) of monoids, however, the
substructures of a group are its submonoids. They need need not
be groups; and even if they are groups, they need not be
subgroups.

In the case of graphs (in the signature
consisting of one binary relation), the induced substructures of a
graph are precisely its induced subgraphs, and its weak
substructures are precisely its subgraphs.

## Substructures as subobjects

For every signature σ, induced substructures of
σ-structures are the subobjects in the concrete
category of σ-structures and
strong homomorphisms (and also in the concrete
category of σ-structures and σ-embeddings).
Weak substructures of σ-structures are the subobjects in the concrete
category of σ-structures and
homomorphisms in the ordinary sense.

## Submodel

In model theory, given a structure M which is a
model of a theory T, a submodel of M in a narrower sense is a
substructure of M which is also a model of T. For example if T is
the theory of abelian groups in the signature (+, 0), then the
submodels of the group of integers (Z, +, 0) are the substructures
which are also groups. Thus the natural numbers (N, +, 0) form a
substructure of (Z, +, 0) which is not a submodel, while the even
numbers (2Z, +, 0) form a submodel which is (a group but) not a
subgroup.

Other examples:

- The algebraic numbers form a submodel of the complex numbers in the theory of algebraically closed fields.
- The rational numbers form a submodel of the real numbers in the theory of fields.
- Every elementary substructure of a model of a theory T also satisfies T; hence it is a submodel.

In the category
of models of a theory and embeddings between them, the
submodels of a model are its subobjects.

## See also

## References

- A Course in Universal Algebra
- Graph Theory
- A shorter model theory

substructure in Chinese: 子模型

# Synonyms, Antonyms and Related Words

base,
basement, basis, bearing wall, bed, bedding, bedrock, bottom, floor, flooring, fond, footing, foundation, fundament, fundamental, ground, grounds, groundwork, hardpan, infrastructure, pavement, principle, radical, riprap, rock bottom, rudiment, seat, sill, solid ground, solid rock,
stereobate, stylobate, substratum, substruction, terra firma,
underbuilding,
undercarriage,
undergirding,
underpinning,
understruction,
understructure