72 Sentences With "underlying set" | Random Sentence Generator

Nevertheless, British security officials were becoming increasingly frustrated with what they viewed as Huawei's failure to fix software flaws in its equipment, particularly discrepancies in the source code – the programs' underlying set of instructions.

The include the following: Among the assets that were part of the original NDS business, Permira is buying Cisco's video security services, including smartcards and software solutions associated with conditional access security; video middleware, or software solutions that provide advanced user experiences on any underlying set-top box hardware; and software for video services.

The identity element can also be written as id. The set G is called the underlying set of the group . Often the group's underlying set G is used as a short name for the group . Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group " or "an element of the underlying set G of the group ".

A Finite topological space is a topological space whose underlying set is finite.

There is no coproduct in Met. The forgetful functor Met → Set assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function. This functor is faithful, and therefore Met is a concrete category.

Pick some mathematical object that has an underlying set, for instance a group, ring, vector space, etc. For any subset of , let be the smallest subobject of that contains , i.e. the subgroup, subring or subspace generated by . For any subobject of , let be the underlying set of .

The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.

In the mathematical theory of probability and measure, a sub-probability measure is a measure that is closely related to probability measures. While probability measures always assign the value 1 to the underlying set, sub- probability measures assign a value lesser than or equal to 1 to the underlying set.

In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator). In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism. A factorization system for a category also gives rise to a notion of embedding.

Also, they are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.

In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements.

Let M be a matroid with an underlying set of elements E, and let N be another matroid on an underlying set F. The direct sum of matroids M and N is the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an independent set of N. The union of M and N is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whose independent sets are those subsets that are the union of an independent set in M and one in N. Usually the term "union" is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the direct sum.

For order one there is only one Latin square with symbol 1 and one quasigroup with underlying set {1}; it is a group, the trivial group.

That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.

The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. pp. 2–3 However not all authors insist on the underlying set of a semigroup being non-empty.

The construction of free groups is a common and illuminating example. Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G: Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let \eta_Y:Y\to GFY be the set map given by "inclusion of generators".

Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlying set. These algebraic free functors have generally the same description as in the detailed description of the free group situation above.

In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

P A Grillet (1995). Semigroups. CRC Press. pp. 3–4 One can logically define a semigroup in which the underlying set S is empty. The binary operation in the semigroup is the empty function from to S. This operation vacuously satisfies the closure and associativity axioms of a semigroup.

This would involve the following: #Defining inverse object functions, checking that they are inverse, and checking that corresponding objects have the same underlying set. #Checking that a set function is "continuous" (i.e., a morphism) in the given category if and only if it is continuous (a morphism) in Top.

In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous.

If is the semidirect product of the normal subgroup and the subgroup , and both and are finite, then the order of equals the product of the orders of and . This follows from the fact that is of the same order as the outer semidirect product of and , whose underlying set is the Cartesian product .

The index is 4. In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint equal-size pieces called cosets. There are two types of cosets: left cosets and right cosets. Cosets (of either type) have the same number of elements (cardinality) as does .

In information theory, the cross-entropy between two probability distributions p and q over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution q, rather than the true distribution p.

There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product. We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category.

This is then an equivalence relation on and the equivalence classes of this relation are the left cosets of . As with any set of equivalence classes, they form a partition of the underlying set. A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal.

In mathematics, a semigroup with no elements (the empty semigroup) is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation.A H Clifford, G B Preston (1964).

In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that B = {0, 1}. Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here.

One of the simplest and most natural examples is the multiset of prime factors of a number . Here the underlying set of elements is the set of prime divisors of . For example, the number 120 has the prime factorization :120 = 2^3 3^1 5^1 which gives the multiset . A related example is the multiset of solutions of an algebraic equation.

In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring.

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space. A measurable space consists of the first two components without a specific measure.

Because Ab has kernels, one can then show that Ab is a complete category. The coproduct in Ab is given by the direct sum; since Ab has cokernels, it follows that Ab is also cocomplete. We have a forgetful functor Ab → Set which assigns to each abelian group the underlying set, and to each group homomorphism the underlying function. This functor is faithful, and therefore Ab is a concrete category.

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure- preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.

Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are epimorphisms in the modern sense. However, this distinction never caught on. It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior.

In general, truth requires a proper fit of elements within the whole system. Very often, though, coherence is taken to imply something more than simple formal coherence. For example, the coherence of the underlying set of concepts is considered to be a critical factor in judging validity. In other words, the set of base concepts in a universe of discourse must form an intelligible paradigm before many theorists consider that the coherence theory of truth is applicable.

Expressed in category-theoretical terms, a set A is Dedekind-finite if in the category of sets, every monomorphism is an isomorphism. A von Neumann regular ring R has the analogous property in the category of (left or right) R-modules if and only if in R, implies . More generally, a Dedekind-finite ring is any ring that satisfies the latter condition. Beware that a ring may be Dedekind-finite even if its underlying set is Dedekind-infinite, e.g.

The infinite symmetric product SP(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xn) with (x1,...,xn,e) and identifying any point with any other point given by permuting its coordinates. In other words its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.

Moreover, given a neighbourhood function N on a Boolean algebra with underlying set B, we can define an interior operator by xI = max {y ∈ B : x ∈ N(y)} thereby obtaining an interior algebra. N(x) will then be precisely the filter of neighbourhoods of x in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions. In terms of neighbourhood functions, the open elements are precisely those elements x such that x ∈ N(x).

In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup. The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality.

If they write about the interaction between music and text, this is purely by instinct: there is no "scholarly consensus" for an underlying set of principles about how the interaction can be optimized. Often, the term "hymnologist" simply refers to anyone who has enough standing within the faith community to be asked to serve on a hymnal committee. Hymnology is sometimes more strictly construed, as in A Dictionary of Hymnology,John Julian, ed., (1902/1985), A Dictionary of Hymnology, Grand Rapids, MI: Kregel.

Examples: (1) For an algebra B and a reduct A of B (that is, an algebra with same underlying set as B but whose set of operations is a subset of the one of B), the canonical (∨, 0)-homomorphism from Conc A to Conc B is weakly distributive. Here, Conc A denotes the (∨, 0)-semilattice of all compact congruences of A. (2) For a convex sublattice K of a lattice L, the canonical (∨, 0)-homomorphism from Conc K to Conc L is weakly distributive.

In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, 6, ...} is a numerical semigroup, the set {0, 1, 3, 5, 6, ...} is not because 1 is in the set and 1 + 1 = 2 is not in the set.

In the area of mathematical logic and computer science known as type theory, a unit type is a type that allows only one value (and thus can hold no information). The carrier (underlying set) associated with a unit type can be any singleton set. There is an isomorphism between any two such sets, so it is customary to talk about the unit type and ignore the details of its value. One may also regard the unit type as the type of 0-tuples, i.e.

For this process, elements of the poset are mapped to (Dedekind-) cuts, which can then be mapped to the underlying posets of arbitrary complete lattices in much the same way as done for sets and free complete (semi-) lattices above. The aforementioned result that free complete lattices do not exist entails that an according free construction from a poset is not possible either. This is easily seen by considering posets with a discrete order, where every element only relates to itself. These are exactly the free posets on an underlying set.

We can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action. Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X. If X has an underlying set, then all definitions and facts stated above can be carried over.

In particular, if, in this case, is a skewfield, then is a field and is a vector space with a bilinear form. An antiautomorphism can also be viewed as an isomorphism , where is the opposite ring of , which has the same underlying set and the same addition, but whose multiplication operation () is defined by , where the product on the right is the product in . It follows from this that a right (left) -module can be turned into a left (right) -module, . Thus, the sesquilinear form can be viewed as a bilinear form .

Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions. To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces Top.

Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Top -> Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. The forgetful functor U has both a left adjoint :D : Set -> Top which equips a given set with the discrete topology, and a right adjoint :I : Set -> Top which equips a given set with the indiscrete topology.

A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a graded Lie algebra is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a chain complex), then it is called a differential graded Lie algebra. A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (so it might be better thought of as a family of Lie algebras).

For any integer k > 0 and any n−dimensional Ck−manifold, the maximal atlas contains a C∞−atlas on the same underlying set by a theorem due to Hassler Whitney. It has also been shown that any maximal Ck−atlas contains some number of distinct maximal C∞−atlases whenever n > 0, although for any pair of these distinct C∞−atlases there exists a C∞−diffeomorphism identifying the two. It follows that there is only one class of smooth structures (modulo pairwise smooth diffeomorphism) over any topological manifold which admits a differentiable structure, i.e. The C∞−, structures in a Ck−manifold.

For coherence theories in general, the assessment of meaning and truth requires a proper fit of elements within a whole system. Very often, though, coherence is taken to imply something more than simple logical consistency; often there is a demand that the propositions in a coherent system lend mutual inferential support to each other. So, for example, the completeness and comprehensiveness of the underlying set of concepts is a critical factor in judging the validity and usefulness of a coherent system.Immanuel Kant, for instance, assembled a controversial but quite coherent system in the early 19th century, whose validity of meaning and usefulness continues to be debated even today.

The outcomes pathway is a set of needed conditions relevant to a given field of action, which are placed diagrammatically in logical relationship to one another and connected with arrows that posit causality. Outcomes along the pathway are also preconditions to outcomes above them. Thus, early outcomes must be in place for intermediate outcomes to be achieved; intermediate outcomes must be in place for the next set of outcomes to be achieved; and so on. An outcomes pathway therefore represents the change logic and its underlying set of assumptions, which are spelled out in the rationales given for why specific connections exist between outcomes and in the theory narrative.

Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively. In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.

In mathematics, the 'extension' of a mathematical concept C is the set that is specified by C. (That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object. The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory.

Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Manp -> Top to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor :U′ : Manp -> Set to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.

The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring. An example of an exponential ring with a nontrivial exponential function is the ring of integers Z equipped with the function E which takes the value +1 at even integers and −1 at odd integers, i.e., the function n \mapsto (-1)^n. This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions.

In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A of finite arity (typically binary operations), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called scalar multiplication between elements of the field (called scalars), and elements of the vector space (called vectors). In the context of universal algebra, the set A with this structure is called an algebra,P.

In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes. In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra.

Walter Seymour Allward's Veritas (Truth) outside Supreme Court of Canada, Ottawa, Ontario Canada For coherence theories in general, truth requires a proper fit of elements within a whole system. Very often, though, coherence is taken to imply something more than simple logical consistency; often there is a demand that the propositions in a coherent system lend mutual inferential support to each other. So, for example, the completeness and comprehensiveness of the underlying set of concepts is a critical factor in judging the validity and usefulness of a coherent system.Immanuel Kant, for instance, assembled a controversial but quite coherent system in the early 19th century, whose validity and usefulness continues to be debated even today.

In topology, a branch of mathematics, the James reduced product or James construction J(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words, its underlying set is the free monoid generated by X (with unit e). It was introduced by . For a connected CW complex X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X. The commutative analogue of the James reduced product is called the infinite symmetric product.

The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor :U : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :F : Set → Ring which assigns to each set X the free ring generated by X. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors :A : Ring → Ab :M : Ring → Mon which "forget" multiplication and addition, respectively.

Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set. An alternate (but equivalent) definition is to expand the structure of a group to define a group as a set equipped with three operations satisfying the same axioms as above, with the "there exists" part removed in the two last axioms, these operations being the group law, as above, which is a binary operation, the inverse operation, which is a unary operation and maps to a^{-1}, and the identity element, which is viewed as a 0-ary operation. As this formulation of the definition avoids existential quantifiers, it is generally preferred for computing with groups and for computer-aided proofs. This formulation exhibits groups as a variety of universal algebra.

Given an n-dimensional formal group law F over R and a commutative R-algebra S, we can form a group F(S) whose underlying set is Nn where N is the set of nilpotent elements of S. The product is given by using F to multiply elements of Nn; the point is that all the formal power series now converge because they are being applied to nilpotent elements, so there are only a finite number of nonzero terms. This makes F into a functor from commutative R-algebras S to groups. We can extend the definition of F(S) to some topological R-algebras. In particular, if S is an inverse limit of discrete R algebras, we can define F(S) to be the inverse limit of the corresponding groups.

In abstract algebra, a partially ordered ring is a ring (A, +, · ), together with a compatible partial order, i.e. a partial order \leq on the underlying set A that is compatible with the ring operations in the sense that it satisfies: :x\leq y implies x + z\leq y + z and :0\leq x and 0\leq y imply that 0\leq x\cdot y for all x, y, z\in A. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring (A, \leq) where A's partially ordered additive group is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, \leq) where \le is additionally a total order.

The topological concept of neighbourhoods can be generalized to interior algebras: An element y of an interior algebra is said to be a neighbourhood of an element x if x ≤ yI. The set of neighbourhoods of x is denoted by N(x) and forms a filter. This leads to another formulation of interior algebras: A neighbourhood function on a Boolean algebra is a mapping N from its underlying set B to its set of filters, such that: #For all x ∈ B, max{y ∈ B : x ∈ N(y)} exists #For all x,y ∈ B, x ∈ N(y) if and only if there is a z ∈ B such that y ≤ z ≤ x and z ∈ N(z). The mapping N of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra.

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G. The set of all permutations of G forms a group under function composition, called the symmetric group on G, and written as Sym(G). Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".

Espen J. Aarseth argues that, although games certainly have plots, characters, and aspects of traditional narratives, these aspects are incidental to gameplay. For example, Aarseth is critical of the widespread attention that narrativists have given to the heroine of the game Tomb Raider, saying that "the dimensions of Lara Croft's body, already analyzed to death by film theorists, are irrelevant to me as a player, because a different-looking body would not make me play differently... When I play, I don't even see her body, but see through it and past it." Simply put, ludologists reject traditional theories of art because they claim that the artistic and socially relevant qualities of a video game are primarily determined by the underlying set of rules, demands, and expectations imposed on the player. While many games rely on emergent principles, video games commonly present simulated story worlds where emergent behavior occurs within the context of the game.

This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through \eta_Y:Y\to GFY via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y. Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let \varepsilon_X:FGX\to X be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each (GX,\varepsilon_X) is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through \varepsilon_X:FGX\to X via a unique set map from Z to GX. This means that (F,G) is an adjoint pair. Hom-set adjunction.

The natural transformations from V' to an arbitrary graph (functor) G constitute the vertices of G while those from E' to G constitute its edges. Although SetC, which we can identify with Grph, is not made concrete by either V' or E' alone, the functor U: Grph -> Set2 sending object G to the pair of sets (Grph(V' ,G), Grph(E' ,G)) and morphism h: G -> H to the pair of functions (Grph(V' ,h), Grph(E' ,h)) is faithful. That is, a morphism of graphs can be understood as a pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of topoi by allowing an object to have multiple underlying sets, that is, to be multisorted.

Given a signature σ, a σ-structure M is a concrete interpretation of the symbols in σ. It consists of an underlying set (often also denoted by "M") together with an interpretation of the function and relation symbols of σ. An interpretation of a constant symbol of σ in M is simply an element of M. More generally, an interpretation of an n-ary function symbol f is a function from Mn to M. Similarly, an interpretation of a relation symbol R is an n-ary relation on M, i.e. a subset of Mn. 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).

The "girth" terminology generalizes the use of girth in graph theory, meaning the length of the shortest cycle in a graph: the girth of a graphic matroid is the same as the girth of its underlying graph.. The girth of other classes of matroids also corresponds to important combinatorial problems. For instance, the girth of a co-graphic matroid (or the cogirth of a graphic matroid) equals the edge connectivity of the underlying graph, the number of edges in a minimum cut of the graph. The girth of a transversal matroid gives the cardinality of a minimum Hall set in a bipartite graph: this is a set of vertices on one side of the bipartition that does not form the set of endpoints of a matching in the graph.. Any set of points in Euclidean space gives rise to a real linear matroid by interpreting the Cartesian coordinates of the points as the vectors of a matroid representation. The girth of the resulting matroid equals one plus the dimension of the space when the underlying set of point is in general position, and is smaller otherwise.