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 ).

* if κ < | M | then N is an elementary substructure of M ;

If N is a substructure of M, one often needs a stronger condition.

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:

It contains an isomorphic copy of M as an elementary substructure.

: 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.

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.

The substructure of a GMC is a complex pattern of filaments, sheets, bubbles, and irregular clumps.

The substructure consisting of a nucleobase plus sugar is termed a nucleoside.

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.

This is an example of optimal substructure.

Such an example is likely to exhibit optimal substructure.

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.

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.

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 >).

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.