THE SEMANTICAL HYPERUNIFICATION PROBLEM

A hypersubstitution of a fixed type τ maps n-ary operation symbols of the type to n-ary terms of the type. Such a mapping induces a unique mapping defined on the set of all terms of type τ . The kernel of this induced mapping is called the kernel of the hypersubstitution, and it is a fully invariant congruence relation on the (absolutely free) term algebra Fτ (X) of the considered type ([2]). If V is a variety of type τ, we consider the composition of the natural homomorphism with the mapping induced by a hypersubstitution. The kernel of this mapping is called the semantical kernel of the hypersubstitution with respect to the given variety. If the pair (s, t) of terms belongs to the semantical kernel of a hypersubstitution, then this hypersubstitution equalizes s and t with respect to the variety. Generalizing the concept of a unifier, we define a semantical hyperunifier for a pair of terms with respect to a variety. The problem of finding a semantical hyperunifier with respect to a given variety for any two terms is then called the semantical hyperunification problem. We prove that the semantical kernel of a hypersubstitution is a fully invariant congruence relation on the absolutely free algebra of ∗Research supported by NSERC of Canada. 176 K. Denecke, J. Koppitz and S.L. Wismath the given type. Using this kernel, we define three relations between sets of hypersubstitutions and sets of varieties and introduce the Galois correspondences induced by these relations. Then we apply these general concepts to varieties of semigroups.


Introduction
Let {f i | i ∈ I} be an indexed set of operation symbols of type τ = (n i ) i∈I , where f i is n i -ary for n i ∈ IN\{0}, and let W τ (X) be the set of all terms built up from elements of the alphabet X = {x 1 , x 2 , . ..} and operation symbols from {f i | i ∈ I}.An arbitrary mapping which preserves the arity, that is, which maps every n i -ary operation symbol of type τ to an n i -ary term of the same type, is called a hypersubstitution of type τ .Any hypersubstitution σ induces a mapping σ : W τ (X) → W τ (X) in the following inductive way: The right hand side of (ii) is the superposition of the term σ(f i ) with the terms σ[t 1 ], . . ., σ[t n i ].This extension is uniquely determined and allows us to define a multiplication, denoted by • h , on the set Hyp(τ ) of all hypersubstitutions of type τ , by where • is the usual composition of functions.This multiplication is associative, and if we denote by σ id the identity hypersubstitution which maps each n i -ary operation symbol f i to the term f i (x 1 , . . ., x n i ), we obtain a monoid (Hyp(τ ); • h , σ id ).
Hypersubstitutions can be used to define the concept of a hyperidentity in a variety V of algebras of type τ.An equation s ≈ t consisting of terms of type τ forms a hyperidentity in V if for all σ ∈ Hyp(τ ) the equations σ[s] ≈ σ[t] are satisfied as identities in V .The identity s ≈ t is called an S-hyperidentity of V , for some subset S ⊆ Hyp(τ ), if σ[s] ≈ σ[t] are identities of V for all σ ∈ S. A variety V is called S-solid if every identity in V is an S-hyperidentity of V.In the special case that S = Hyp(τ ), we speak of a hyperidentity of the variety V and of a solid variety.For more information on hyperidentities and solidity, we refer the reader to [4].
If σ is a hypersubstitution of type τ , it is very natural to ask for its kernel, By definition, the kernel of a hypersubstitution is an equivalence relation on the set W τ (X).In fact this kernel turns out to be a fully invariant congruence.
Proposition 1. 1 ([4]).The kernel of a hypersubstitution σ of type τ is a fully invariant congruence relation on the absolutely free algebra F τ (X) = (W τ (X); (f i ) i∈I ) (where the operations f i are defined by Let V be an arbitrary variety of algebras of type τ and let IdV be the set of all identities satisfied in V .Then we generalize the concept of a kernel of a hypersubstitution of type τ in the following way: Definition 1.2.The set will be called the kernel of σ with respect to V or the semantical kernel of σ.The kernel kerσ of a hypersubstitution σ will be called the syntactical kernel. It is well known that the variety Alg(τ ) of all algebras of type τ has the property that an identity s ≈ t holds in it iff s = t.This means that the syntactical kernel is in fact the semantical kernel with respect to the variety Alg(τ ).
For the solution of the word problem in a variety V , the concept of a unifier is important.A substitution is any mapping s : X −→ W τ (X), and any such substitution has a unique extension s : W τ (X) −→ W τ (X).
A substitution s for which s(t) ≈ s(t ) is an identity in V is called a unifier for t and t with respect to the variety V .If V is the variety of all algebras of type τ, this happens only if s(t) = s(t ), and in this case we call s a syntactical unifier for t and t .To solve the unification problem (see [6]) means to decide whether for two given terms there exists a unifier or not.The concept of a unifier can be generalized to the concept of a hyperunifier and the unification problem to the hyperunification problem, by considering hypersubstitutions instead of substitutions.
When such a hyperunifier exists, the terms t and t are called hyperunifiable in the variety V .If V is the variety Alg(τ ) of all algebras of type τ , the unifier is called a syntactical hyperunifier, otherwise it is called a semantical hyperunifier.
The semantical hyperunification problem for a variety V is then the problem of deciding, for any two distinct terms, whether the terms are hyperunifiable in V or not.Since by definition our ker V σ is the set of all pairs of terms for which σ is a semantical hyperunifier with respect to the variety V , we can use such kernels in solving the semantical hyperunification problem.
When a hyperunifier exists for two terms, we want to compare all such hyperunifiers.In order to do this, we use the product • h to define a binary relation on Hyp(τ ): Definition 1.4.Let σ 1 and σ 2 be hypersubstitutions of type τ .Then σ 1 σ 2 if there is a hypersubstitution λ of type τ such that Clearly, the relation is reflexive (using the identity hypersubstitution σ id ) and transitive (since the product of two hypersubstitutions of type τ is again a hypersubstitution of type τ ).The intersection of a reflexive and transitive relation (that is, a quasiorder) with its inverse relation gives an equivalence relation on Hyp(τ ), defined by This relation is the well-known Green's relation L, which is a right congruence on the monoid Hyp(τ ) (see [3]).Now we can define a relation ≤ on the quotient set Hyp(τ )/ by setting

The semantical hyperunification problem
It is well-known and easy to check that this definition gives an order relation on Hyp(τ )/ .
Using the quasiorder on the set Hyp(τ ) one can also generalize the concept of a most general unifier of the terms t and t to the concept of a most general hyperunifier of t and t : Definition 1.5.Let t and t be two terms of type τ and let σ 1 and σ 2 be two hyperunifiers of t and t .Then σ 1 is more general than σ 2 if σ 1 σ 2 .A hyperunifier σ of t and t is called a most general (or minimal) hyperunifier of t and t if σ σ for all hyperunifiers σ of t and t .

The semantical kernel of a hypersubstitution
The syntactical kernel is the (semantical) kernel with respect to the variety Alg(τ ) of all algebras of type τ .The semantical and syntactical kernels are closely related to each other.Let F τ (X) be the absolutely free algebra of type τ and let F V (X) be the relatively free algebra with respect to the variety V of type τ .We denote by natIdV the natural homomorphism which maps each term t of type τ to the class [t] IdV .Then we have: Proposition 2.1.Let V be a variety and σ a hypersubstitution, both of type τ .Then ker V σ = ker(natIdV • σ).
P roof.For any terms t and t , we have Since the composition natIdV • σ is not a homomorphism from F τ (X) to F V (X), we could not use for the proof the fact that the kernel of a homomorphism is a congruence relation.Moreover, the kernels of semantical hypersubstitutions are fully invariant congruence relations.To prove this, we will use the fact that any substitution s : X −→ W τ (X) can be uniquely extended to an endomorphism s : Using induction on the complexity of the term t = f (r 1 , . . .r n ), it can be shown that this last equation is valid for arbitrary terms as well as for the operation symbol f i , so that s(t(t 1 , . . ., t n i )) = t(s(t 1 ), . . ., s(t n i )) for every term t.Here t(t 1 , . . ., t n i ) means the composition (superposition) of terms.
Theorem 2.3.Let σ be a hypersubstitution of type τ = (n i ) i∈I , with n i ≥ 1 for all i ∈ I. Then ker V σ is a fully invariant congruence relation on the absolutely free algebra F τ (X).P roof.By Proposition 2.2 we only have to show that ker V σ is fully invariant.Let s : X −→ W τ (X) be a substitution and let s be its extension.Consider a mapping s * : X −→ W τ (X) defined by s * (x) := σ[s(x)] for every x ∈ X.Since s * is also a substitution, it can be uniquely extended to an endomorphism s * : We show by induction on the complexity of a term t that The semantical hyperunification problem Now let (t, t ) ∈ ker V σ, and let s be any substitution.Then σ This means that (s(t), s(t )) ∈ ker V σ, and hence ker V σ is fully invariant.
Since ker V σ is fully invariant, it is an equational theory and therefore there is a variety V of type τ for which ker V σ = IdV .It is natural then to compare the varieties V and V , or dually the sets of identities IdV and ker V σ.We will consider the possibilities that ker V σ = IdV , that ker V σ ⊆ IdV and that ker V σ ⊇ IdV .These possibilities define three relations and three Galois correspondences which we will study in the next section.

Three Galois correspondences
Let W be a given variety of type τ .We will denote by L(W ) the subvariety lattice of W .In this section we define and study three relations KER, R and R between Hyp(τ ) and L(W ), based on the relationship between ker V σ and IdV .We also study the three Galois correspondences, between sets of hypersubstitutions and collections of varieties, induced by these three relations.Note that in order to compare kernels and sets of identities of the form IdV , we shall regard IdV as a set consisting of pairs of terms of type τ , by identifying the identity p ≈ q with the pair (p, q) ∈ W τ (X) 2 .
Definition 3.1.Let KER ⊆ Hyp(τ ) × L(W ) be the relation defined by In [1] the binary relation R ⊆ Hyp(τ ) × L(W ) defined by was considered.The set σ[V ] is defined as the set of all algebras which are derived from algebras from V , using the hypersubstitution σ.
) is an algebra, then the algebra σ(A) = (A; σ(f i ) i∈I ) of the same type and having the same universe A is called a derived algebra of A. Hy- For more background see [4]. When ∈ IdV , using the so-called "conjugate property" (see [4]).But this means IdV ⊆ ker V σ.Conversely, from IdV ⊆ ker V σ, we get σ[V ] ⊆ V .Altogether we see that the relation R ⊆ Hyp(τ ) × L(W ) can also be defined by We define a third relation by Directly from the definitions, we obtain Each of these three relations defines a Galois correspondence between the sets Hyp(τ ) and L(W ).Let S be a subset of Hyp(τ ) and let L be a subset of L(W ).In [1], the Galois correspondence induced by the relation R was defined as the pair (η, θ) with The following result was proved in [1].
).For any subset L of L(W ) and for any subset S of Hyp(τ ), the image θ(L) is a submonoid of Hyp(τ ) and the image η(S) is a sublattice of L(W ).
It is easy to see that the lattice η(S) is in fact a complete sublattice of L(W ).We want to check now whether images under β and δ are also submonoids of Hyp(τ ), and whether images under α and γ are also sublattices of L(W ).
The following consequence of the fact that KER = R ∩ R will be useful.
Proposition 3.4.For every subset S ⊆ Hyp(τ ) and every subset L ⊆ L(W ), the operators γ, η, α, δ, θ and β satisfy For a singleton set {V } consisting of one variety, the monoid θ({V }) is just the monoid of V -proper hypersubstitutions, usually denoted by P (V ) (see [5]).P lonka also defined another monoid P 0 (V ) associated with V , the monoid of all inner hypersubstitutions.We will show that for single varieties our other two images, δ({V }) and β({V }), are also monoids, and in analogy with P (V ) we will call them P 1 (V ) and P 2 (V ), respectively.
P roof.From Proposition 3.4, we get P 1 (V ) = P (V ) ∩ P 2 (V ).Since P (V ) is a submonoid of Hyp(τ ) by Proposition 3.3, we only have to show that P 2 (V ) is a submonoid of Hyp(τ ).Let σ 1 and σ 2 be in P 2 (V ), so that (σ 1 , V ), (σ 2 , V ) ∈ R .Then ker V σ 1 ⊆ IdV and ker V σ 2 ⊆ IdV .Therefore for every pair (s, t) of terms of type τ we have: Now our claim holds for arbitrary subsets L of L(W ), since β(L) is the intersection of β({V }) for all V ∈ L, and similarly for δ.This shows that for all three maps, any images under the maps are submonoids of Hyp(τ ).Now we turn to the dual maps α and γ, to see if their images are always sublattices.The following lemma will help us to answer this question.
Lemma 3.6.Let σ be a hypersubstitution of type τ and let V 1 , V 2 be varieties of type τ .Then P roof.Using the definition and the fact that Let Con inv F τ (X) be the lattice of all fully invariant congruence relations on the absolutely free algebra F τ (X) of type τ .Since the kernels of hypersubstitutions are elements of the lattice Con inv F τ (X), we may consider a mapping ϕ : L(Alg(τ )) × Hyp(τ ) −→ Con inv F τ (X), which associates to each hypersubstitution σ and to each variety V of the type τ the kernel ker V σ.If we fix the hypersubstitution σ, the resulting mapping on L(Alg(τ )) is anti-isotone.
Corollary 3.7.If V 1 ⊆ V 2 are varieties of type τ and if σ is a hypersubstitution of the same type, then ker , so by the previous Lemma we get Let T R be the trivial variety of type τ .It is easy to see that for any hypersubstitution σ, we have ker T R σ equal to the set W τ (X) 2 of all identities of type τ , and equal to IdT R. Since T R ⊆ V ⊆ Alg(τ ) for any variety V of type τ , we have ker Alg(τ ) σ = kerσ ⊆ ker V σ ⊆ ker T R σ.This means that the syntactical kernel of a hypersubstitution is always a subset of any semantical kernel of that hypersubstitution.The mapping ϕ defined above is also surjective: for any fully invariant congruence relation Σ of F τ (X), there is a variety V such that Σ = IdV and a hypersubstitution, namely σ id , for which IdV = ker V σ id .Now we can prove the following.Theorem 3.8.Let W be a variety of type τ .Then for any subset S of Hyp(τ ), the images α(S) and γ(S) are complete join-subsemilattices of L(W ).
. The same argument extends to any family of varieties in L(W ), showing that α(S) is closed under arbitrary joins of varieties in L(W ).The same proof, but with set inclusion replaced by equality, holds for γ(S).(But the result for γ(S) is also a consequence of the result for α and Proposition 3.4.) We now present a counterexample showing that γ(S) (and hence by Proposition 3.4 also α(S)) is not in general a sublattice of L(W ).We consider the type τ = (2), the two-element alphabet X 2 = {x 1 , x 2 } and the hypersubstitution σ which maps the binary operation symbol f to the term f (x 1 , f (x 1 , x 2 )).Instead of f (x 1 , x 2 ) we will write x 1 x 2 .We denote by σ the submonoid of Hyp(2) generated by σ.If Σ is a set of equations of type (2), we set χ σ [Σ] to be the set {σ Clearly, classes of the form M odχ σ [Σ] are σ -solid varieties.
We define two varieties U 1 and U 2 of type (2), by We need the following preliminary Lemma.Lemma 3.9.With σ, U 1 and U 2 as defined above, P roof.Since U 1 and U 2 are σ -solid, we have only to show that ker U i σ ⊆ IdU i for i = 1, 2. We will do this only for U 1 , since the proof for U 2 is similar.We make the following observations regarding identities of U 1 : 2 Let W(X) denote the set of all terms of type (2), then let σ[W (X)] be the set of all σ[w] with w ∈ W (X). We show by induction on the complexity of the term w that for all ψ : X −→ W (X) and all w ∈ W (X), and all x 1 ∈ X, Here ψ is the unique extension of ψ to the set W (X). First, if w Inductively, assume that w = f (w 1 , w 2 ) and that w 1 , w 2 satisfy the implication.Then ψ 3 Because of 2, we can modify the substitution rule in our case to

This means we do not have to consider substitutions which map variables to terms outside of σ[W (X)]. 4 Since each identity σ[s]
], the compatibility rule can be modified as follows: Now we show by induction on the length of a derivation that the following propositions are satisfied: We check each of the five derivation rules in turn.
Substitution rule: Here we use 2. Assume that σ[s] ≈ σ[t] ∈ IdU with s ≈ t ∈ IdU 1 and consider two substitutions ϕ : X −→ σ[W (X)] and ϕ * : X −→ W (X) with ϕ(x 1 ) = σ[ϕ * (x 1 )], for x 1 ∈ X.We show by induction on the complexity of the term w ∈ W (X) that Here φ and φ * are the unique extensions of ϕ and ϕ * , resp., to the set W (X). Inductively, suppose that w = f (w 1 , w 2 ) and that φ(σ Now we want to calculate the images α(S), η(S), and γ(S) when S is the largest or smallest possible monoid of hypersubstitutions.We will take W to be the largest variety Alg(τ ).It is easy to see that for every variety V of type τ we have ker V σ id = IdV .This means that the image of the smallest submonoid S = {σ id } under all three maps α, η and γ is all of Alg(τ ).The next two lemmas investigate the images of the largest monoid S = Hyp(τ ).Lemma 3.11.Let τ = (n i ) i∈I be a type with n i ≥ 2 for some i ∈ I. Then α(Hyp(τ )) = {T R}. (Here T R is the trivial variety of type τ .)P roof.Since the trivial variety T R satisfies all possible identities of type τ , we have ker T R σ = IdT R for all hypersubstitutions σ in Hyp(τ ).This gives T R ∈ α(Hyp(τ )), i.e. {T R} ⊆ α(Hyp(τ )).
Next we consider the remaining case, where our type contains only unary operation symbols.Lemma 3.12.Let τ = (n i ) i∈I with n i = 1 for all i ∈ I. Then α(Hyp(τ )) = {T R, B}, where B = M od{f i (x 1 ) ≈ x 1 | i ∈ I}.P roof.As before, we have T R ∈ α(Hyp(τ )), but now we show that also B ∈ α(Hyp(τ )).It is easy to see that any term over B contains exactly one variable, and for two terms s and t we have s ≈ t ∈ IdB iff the variable in s is the same as the variable in t.It follows from this that for any hypersubstitution σ, σ[s] ≈ σ[t] is in IdB iff s ≈ t is in IdB.This makes ker B σ = IdB for all σ, so B is in α(Hyp(τ )).
We now have {T R, B} ⊆ α(Hyp(τ )).For the opposite inclusion, we let V be a variety of type τ for which ker V σ ⊆ IdV for all σ ∈ Hyp(τ ).Let σ be a hypersubstitution with σ(f i ) = x 1 for all i ∈ I. Then f i (x 1 ) ≈ x 1 ∈ ker V σ for all i ∈ I, so by our assumption on V we have f i (x 1 ) ≈ x 1 ∈ IdV for all i ∈ I.This shows that V ⊆ B. It is well-known that B has no subvarieties other than B and T R, so we have V ∈ {T R, B}.Thus α(Hyp(τ )) ⊆ {T R, B}, and altogether we have α(Hyp(τ )) = {T R, B}.
V ∈ η(Hyp(τ )) means that IdV ⊆ ker V σ for all σ ∈ Hyp(τ ), that is, that for any hypersubstitution σ and any s ≈ t ∈ IdV , the identity σ[s] ≈ σ[t] also holds in V .This says precisely that V is solid, so that the image under η of the monoid Hyp(τ ) is the lattice S(τ ) of all solid varieties of type τ .
Since γ is the intersection of η and α, we can also determine the image γ(Hyp(τ )).Since the trivial variety T R and the variety B considered in Lemma 3.12 are solid, we see that γ(Hyp(τ )) is either {T R}, if there is an operation symbol f j with n j ≥ 2, or {T R, B} otherwise.or For the operator η we have The semantical hyperunification problem 191

Relations on sets of hypersubstitutions
In this section we show that to calculate the images of the operators introduced in the previous section we can restrict our efforts to certain "special" hypersubstitutions.This is also the case if we want to test whether an identity s ≈ t is a hyperidentity in the variety V .Then we can restrict our checking to a subset of the given set of hypersubstitutions.In [5] J. P lonka introduced the following equivalence relation on Hyp(τ ): Let V be a variety of type τ .Two hypersubstitutions σ 1 and σ 2 of type τ are called V -equivalent, and we write σ 1 ∼ V σ 2 , if for all operation symbols f i of the type the identities σ The following lemma shows how the relation ∼ V can be used.
Then the following two conditions (i) and (ii) are equivalent: Moreover, (iii) For all s, t ∈ W τ (X), and for all σ 1 , σ 2 ∈ Hyp(τ From the last condition it follows that the monoid P (V ) of all proper hypersubstitutions of the variety V is the union of equivalence classes with respect to ∼ V .This means that to test if a given identity s ≈ t is a hyperidentity of V , we need only check the application of σ for one representative σ from each ∼ V equivalence class.
It is easy to see that V -equivalent hypersubstitutions induce equal kernels with respect to V .Proposition 4.2.Let V be a variety of type τ and let σ
Although V -equivalent hypersubstitutions produce the same kernels with respect to V , the converse is not always true: it is possible for ker V σ 1 = ker V σ 2 when σ 1 and σ 2 are not V -equivalent.As an example we consider the semigroup variety V = M od{x 1 (x 2 x 3 ) ≈ (x 1 x 2 )x 3 , x 1 x 2 x 3 ≈ x 1 x 3 } and the hypersubstitutions σ x 1 and σ x 2 1 which map the binary operation symbol to the terms x 1 and x 2 1 , respectively.Then σ x 1 ∼ V σ x 2 1 since the idempotent identity x 2  1 ≈ x 1 is not satisfied in V ; but 1 .Therefore, it makes sense to define the following relation on Hyp(τ ): Definition 4.3.Let V be a variety of type τ .Let ∼ ker V be the relation on Hyp(τ ) defined by By Proposition 4.2, we see that ∼ V ⊆∼ ker V (although these relations need not be equal), and that P (V ), P 1 (V ), P 2 (V ) are unions of equivalence classes with respect to ∼ ker V .It is easy to see that neither ∼ V nor ∼ ker V is a congruence relation on Hyp(τ ).
Another consequence of Proposition 4.2 is that if σ 1 ∼ V σ 2 and σ 1 is a hyperunifier with respect to V of the terms t and t , then σ 2 is also a hyperunifier with respect to V of t and t .This leads us to compare the relation ∼ ker V with the relation (the Green's relation L) introduced in Section 1.As a consequence of this, we have: So, kerσ 1 = kerσ 2 .
In the semantical case, when V ⊂ Alg(τ ), Lemma 4.4 no longer holds without an additional restriction on V .Lemma 4.4 is a special case of the following.
If V is a solid variety, then the inclusion of Lemma 4.6 holds for all hypersubstitutions, giving the following special case.Corollary 4.7.When V is a solid variety of type τ , then σ ρ =⇒ ker V ρ = ker V σ.
Altogether for solid varieties V the inclusion ⊆ ∼ ker V is satisfied, and this means that if σ 1 σ 2 and σ 1 is a hyperunifier with respect to V of the terms t and t , then σ 2 is also a hyperunifier with respect to V of t and t .