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A substructure N of M is elementary if and only if it passes the Tarski – Vaught test: Every first-order formula φ ( x, b < sub > 1 </ sub >, …, b < sub > n </ sub >) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementary equivalent with the Ehrenfeucht – Fraïssé games.

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## Some Related Sentences

substructure and N

**A**

__substructure__

**of**

**a**σ-structure

**M**

**is**obtained by taking

**a**subset

__N__

**of**

**M**which

**is**closed under

**the**interpretations

**of**all

**the**function symbols

**in**σ

**(**hence includes

**the**interpretations

**of**all constant symbols

**in**σ ),

**and**then restricting

**the**interpretations

**of**

**the**relation symbols to

__N__

**.**An

**elementary**

__substructure__

**is**

**a**very special case

**of**this ;

**in**particular an

**elementary**

__substructure__satisfies exactly

**the**same

**first-order**sentences as

**the**original structure

**(**its

**elementary**extension ).

In this case

__N__**is**called an**elementary**__substructure__**of****M****if**every**first-order**σ-formula**φ****(****a****<****sub****>****1****</****sub****>,****…,****a****<****sub****>****n****</****sub****>)****with****parameters****a****<****sub****>****1****</****sub****>,****…,****a****<****sub****>****n****</****sub****>**from__N__**is**true**in**__N__**if****and****only****if****it****is**true**in****M****.**
If

__N__**is**an**elementary**__substructure__**of****M****,****M****is**called an**elementary**extension**of**__N__**.**An embedding h**:**__N__→**M****is**called an**elementary**embedding**of**__N__into**M****if**h**(**__N__)**is**an**elementary**__substructure__**of****M****.**__N__

**is**an

**elementary**

__substructure__

**of**

**M**

**if**

__N__

**and**

**M**

**are**

**structures**

**of**

**the**same signature σ such

**that**for all

**first-order**σ-formulas

**φ**

**(**

**x**

**<**

**sub**

**>**

**1**

**</**

**sub**

**>,**

**…,**

**x**

**<**

**sub**

**>**

**n**

**</**

**sub**

**>)**

**with**free variables

**x**

**<**

**sub**

**>**

**1**

**</**

**sub**

**>,**

**…,**

**x**

**<**

**sub**

**>**

**n**

**</**

**sub**

**>,**

**and**all elements

**a**

**<**

**sub**

**>**

**1**

**</**

**sub**

**>,**

**…,**

**a**

**<**

**sub**

**>**

**n**

**</**

**sub**

**>**

**of**

__N__

**,**

**φ**

**(**

**a**

**<**

**sub**

**>**

**1**

**</**

**sub**

**>,**

**…,**

**a**

**<**

**sub**

**>**

**n**

**</**

**sub**

**>)**holds

**in**

__N__

**if**

**and**

**only**

**if**

**it**holds

**in**

**M**

**:**

substructure and M

**:**

**Every**countable theory which

**is**satisfiable

**in**

**a**model

__M__

**,**

**is**satisfiable

**in**

**a**countable

__substructure__

**of**

__M__

**.**

Given

**a**model__M__**of****a**Skolem theory T**,****the**smallest__substructure__containing**a**certain set**A****is**called**the**Skolem hull**of****A****.**

substructure and is

In particle physics

**,**an**elementary**particle or fundamental particle__is__**a**particle not known to have__substructure__**,**thus**it**__is__not known to be made up**of**smaller particles**.**
If an

**elementary**particle truly**has**no__substructure__**,**then**it**__is__one**of****the**basic building blocks**of****the**universe from which all other particles**are**made**.**
Representation

**of**an organic compound | organic hydroxyl group**,**where R represents**a**hydrocarbon or other organic moiety**,****the**red**and**grey spheres represent oxygen**and**hydrogen atoms**,**respectively**,****and****the**rod-like connections between these**,**covalent chemical bond s**.****A**hydroxyl__is__**a**chemical functional group containing an oxygen atom connected by**a**covalent bond to**a**hydrogen atom**,****a**pairing**that****can**be simply understood as**a**__substructure__**of****the**water molecule**.**
As

**a**result**,**warhead components**are**contained within an aluminium honeycomb__substructure__**,**sheathed**in**pyrolytic graphite-epoxy resin composite**,****with****a**heat-shield layer on top which__is__constructed out**of**3-Dimensional Quartz Phenolic**.**
By

**the**first isomorphism theorem**,****the**image**of****A**under ƒ__is__**a**__substructure__**of**B isomorphic to**the**quotient**of****A**by this congruence**.**
Furthermore

**,****the**pyramid__substructure____is__reminiscent**of****the**plan**of**Khasekhemwy ’ s mud-brick funerary enclosure at Abydos**.**
The

__substructure__**of****the**South Tomb__is__entered through**a**tunnel-like corridor**with****a**staircase**that**descends about 30m before opening up into**the**pink granite burial chamber**.**
It

__is__applicable to problems exhibiting**the**properties**of**overlapping subproblems which**are****only**slightly smaller**and**optimal__substructure__**(**described below ).
Finding

**the**shortest path**in****a**graph using optimal__substructure__;**a**straight line indicates**a**single edge ;**a**wavy line indicates**a**shortest path between**the****two**vertices**it**connects**(**other nodes on these paths**are**not shown );**the**bold line__is__**the**overall shortest path from start to goal**.**
Likewise

**,****in**computer science**,****a**problem**that****can**be broken down recursively__is__said to have optimal__substructure__**.**
Consequently

**,****the**first step towards devising**a**dynamic programming**solution**__is__to check whether**the**problem exhibits such optimal__substructure__**.**
For example

**,**given**a**graph G =( V**,**E ),**the**shortest path p from**a**vertex u to**a**vertex v exhibits optimal__substructure__**:**take any intermediate vertex w on this shortest path p**.**If p__is__truly**the**shortest path**,**then**the**path p**<****sub****>****1****</****sub****>**from u to w**and**p**<****sub****>**2**</****sub****>**from w to v**are**indeed**the**shortest paths between**the**corresponding vertices**(**by**the**simple cut-and-paste argument described**in**CLRS ).
Occasionally

**,****the**reason behind such Ramsey-type results__is__**that****the**largest partition class always contains**the**desired__substructure__**.**
The Ara Pacis

__is__seen to embody without conscious effort**the**deep-rooted ideological connections among cosmic sovereignty**,**military force**and**fertility**that**were first outlined by Georges Dumézil**,**connections which**are**attested**in**early Roman culture**and**more broadly**in****the**__substructure__**of**Indo-European culture at large**.**
In computer science

**,****a**problem__is__said to have optimal__substructure__**if**an optimal**solution****can**be constructed efficiently from optimal solutions**of**its subproblems**.**
Typically

**,****a**greedy algorithm__is__used to solve**a**problem**with**optimal__substructure__**if****it****can**be proved by induction**that**this__is__optimal at each step**(**Cormen et al**.**
As an example

**of****a**problem**that**__is__unlikely to exhibit optimal__substructure__**,**consider**the**problem**of**finding**the**cheapest airline ticket from Buenos Aires to Moscow**.**
If minimizing

**the**local functions__is__**a**problem**of**" lower order ",**and****(**specifically )**if****,**after**a**finite number**of**these reductions**,****the**problem becomes trivial**,**then**the**problem**has**an optimal__substructure__**.**
Firstly

**,****a**cardinal κ__is__inaccessible**if****and****only****if**κ**has****the**following reflection property**:**for all subsets U ⊂ V**<****sub****>**κ**</****sub****>,**there exists α**<**κ such**that**__is__an**elementary**__substructure__**of****.**

substructure and elementary

It

**is**provable**in**ZF**that**∞ satisfies**a**somewhat weaker reflection property**,**where**the**__substructure__**(**V**<****sub****>**α**</****sub****>,**∈, U ∩ V**<****sub****>**α**</****sub****>)****is****only**required to be '__elementary__'**with**respect to**a**finite set**of**formulas**.**
Iterating countably many times results

**in****a**closure operator Taking an arbitrary subset such**that****,****and**having defined one**can**see**that****also****is**an__elementary____substructure__**of**by**the****Tarski****–****Vaught****test****.****Every**Skolem theory

**is**model complete

**,**i

**.**e

**.**every

__substructure__

**of**

**a**model

**is**an

__elementary__

__substructure__

**.**

substructure and if

If

**N****is****a**__substructure__**of****M****,**then both**N****and****M****can**be interpreted as**structures****in****the**signature σ**<****sub****>****N****</****sub****>**consisting**of**σ together**with****a**new constant symbol for every element**of****N****.****N****is**an**elementary**__substructure__**of****M**__if__**and****only**__if__**N****is****a**__substructure__**of****M****and****N****and****M****are**elementarily**equivalent**as σ**<****sub****>****N****</****sub**>-**structures****.**
Let

**M**be**a**structure**of**signature σ**and****N****a**__substructure__**of****M****.****N****is**an**elementary**__substructure__**of****M**__if__**and****only**__if__for every**first-order****formula****φ****(****x****,**y**<****sub****>****1****</****sub****>,****…,**y**<****sub****>****n****</****sub****>)**over σ**and**all elements**b****<****sub****>****1****</****sub****>,****…,****b****<****sub****>****n****</****sub****>**from**N****,**__if__**M****x****φ****(****x****,****b****<****sub****>****1****</****sub****>,****…,****b****<****sub****>****n****</****sub**>), then there**is**an element**a****in****N**such**that****M****φ****(****a****,****b****<****sub****>****1****</****sub****>,****…,****b****<****sub****>****n****</****sub**>).

substructure and only

Beam bridges

**are**horizontal beams supported at each end by__substructure__units**and****can**be either simply supported**when****the**beams__only__connect across**a**single span**,**or continuous**when****the**beams**are**connected across**two**or more spans**.**
Land clearance

**and**excavation**of****the**foundation began**in**March 1965**,**but delays**in**obtaining congressional funding meant**that**__only__**the**three-story__substructure__was complete by 1970**.**
By late September 1954

**,**73 percent**of****the**superstructure had been completed**and**__only__stone protections for**the**piers remained to be finished for**the**__substructure__**.****A**

**first-order**theory T

**has**quantifier elimination

**if**

**and**

__only__

**if**for any

**two**models B

**and**C

**of**T

**and**for any common

__substructure__

**A**

**of**B

**and**C

**,**B

**and**C

**are**elementarily

**equivalent**

**in**

**the**language

**of**T augmented

**with**constants from

**A**

**.**

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