homomorphism
homomorphism A structurepreserving mapping between algebras. A homomorphism allows the modeling, simulation, or representation of the structure of one algebra within another, possibly in a limited form. Let A and B be algebras and h a function from A to B. Suppose that A contains an nary operation f_{A}, while B contains a corresponding operation f_{B}. If h is a homomorphism it must satisfy h(f_{A}(a_{1},…,a_{k})) = f_{B}(h(a_{1}),…,h(a_{k}))
for all elements a_{1},…,a_{k} of A and every “corresponding” pair of operations of A and B.
The idea that f_{A} and f_{B} are “corresponding” operations is made precise by saying that A and B are algebras over the same signature Σ, while f is an operation symbol in Σ with which A and B associate the operations f_{A} and f_{B} respectively. A homomorphism from A to B is any function h from A to B that satisfies the condition given above for each f in Σ. As applications of this idea, the semantic functions involved in denotational semantics can be viewed as homomorphisms from algebras of syntax to algebras of semantic objects. Usually, to define a semantic function by induction on terms is to define a homomorphism on a term algebra. In several important cases, compilers can be designed as homomorphisms between two algebras of programs.
Special cases of this general definition occur when A and B belong to one of the familiar classes of algebraic structures. For example, let A and B be monoids, with binary operations ◦_{A} and ◦_{B} and identity elements e_{A} and e_{B}. Then, rewriting the general condition above, a homomorphism from A to B satisfies h(x ◦_{A} y) = h(x) ◦_{B} h(y) h(e_{A}) = e_{B}
A further specialization from formal language theory arises with monoids of words, where the binary operation is concatenation and the nullary operation is the empty word. Let S and T be alphabets, and let h be a function from S to T*, i.e. a function that gives a Tword for each symbol in S. Then h can be extended to Swords, by concatenating its values on individual symbols: h(s_{1},…,s_{n}) = h(s_{1}),…,h(s_{n})
This extension of h gives a monoid homomorphism from S* to T*. Such an h is said to be Λfree if it gives a nonempty Tword for each symbol in S.
h can be further extended to a mapping on languages, giving, for any subset L of S*, its homomorphic image h(L): h(L) = {h(w)  w ∈ L}
Similarly the inverse homomorphic image of L ⊆ T* is h^{–1}(L) = {w  h(w) ∈ L}
These languagemappings are also homomorphisms, between the monoids of languages over S and over T, the binary operation being concatenation of languages.
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