Complexity of Hypersubstitutions and Lattices of Varieties

Hypersubstitutions are mappings which map operation symbols to terms. The set of all hypersubstitutions of a given type forms a monoid with respect to the composition of operations. Together with a second binary operation, to be written as addition, the set of all hypersubsti-tutions of a given type forms a left-seminearring. Monoids and left-seminearrings of hypersubstitutions can be used to describe complete sublattices of the lattice of all varieties of algebras of a given type. The complexity of a hypersubstitution can be measured by the complexity of the resulting terms. We prove that the set of all hypersubstitutions with a complexity greater than a given natural number forms a sub-left-seminearring of the left-seminearring of all hypersubstitutions of the considered type. Next we look to a special complexity measure, the operation symbol count op(t) of a term t and determine the greatest M-solid variety of semigroups where M = H op 2 is the left-seminearring of all hypersubstitutions for which the number of operation symbols occurring in the resulting term is greater than or equal to 2. For every n ≥ 1 and for M = H op n we determine the complete lattices of all M-solid varieties of semigroups.


Introduction
Let τ = (n i ) i∈I be a type indexed by a set I, with operation symbols f i of arity n i for n i ∈ IN.Let X = {x 1 , x 2 , . ..} be a countably infinite set of variables.We denote by W τ (X) the set of all terms of type τ over the alphabet X.
The complexity of a term or a tree plays an important role in Computer Science applications and can be measured in different ways.One can count the number of occurrences of variables in the tree, or one can also count the number of operation symbols.Another method is to compare the lengths of all paths from the root to the leaves in the tree.The length of the longest such path gives the depth while the length of the shortest such path gives the mindepth ( [4]).In [7] a general complexity function, called a valuation Complexity of hypersubstitutions and lattices of varieties 33 of terms of type τ into an algebra IN τ of type τ with the set of all natural numbers IN as universe, was considered.
Definition 1.1 [7].Let a be a fixed natural number and let ) be an algebra of type τ with base set IN. Let v : X → IN be a mapping defined by v(x) = a for all x ∈ X.Then v has a unique extension (which we also denote by v) to the set W τ (X) of all terms which is a homomorphism from the free algebra F τ (X) = (W τ (X); ( fi ) i∈I ) into IN τ .This extension homomorphism is called a valuation of terms of type τ into IN τ if it satisfies the condition v(t) ≥ v(x) for every variable x and every term t.The algebra IN τ will be called the valuation algebra of the valuation v, and a is called the base value of the valuation.Now we want to give some examples of such valuations of terms.The operation symbol count of a term, denoted by op(t) is inductively defined by In this case the operations The minimum depth of a term t, denoted by mindepth(t), is the length of the shortest path from the root to a vertex in the tree, and is defined inductively by For mindepth we use the operations f The depth of a term t, denoted by depth(t), is inductively defined by Th. Changphas and K. Denecke In all our examples the operations f IN i of the valuation algebra are monotone, meaning that the following condition (OC) is satisfied: (OC) If a j ≤ b j for 1 ≤ j ≤ n i and f i is an n i -ary operation symbol of type τ , then for the corresponding operation We denote the superposition of the term s with the terms t 1 , . . ., t n by s(t 1 , . . ., t n ).It was proved in [7] that for valuations satisfiying (OC) the following condition is satisfied.Lemma 1.2 .Let v be a valuation of terms of type τ into IN τ which satisfies (OC).Then for any n-ary term s and m-ary terms t 1 , . . ., t n we have v(s(t 1 , . . ., t n )) ≥ v(s).
Hypersubstitutions are defined as mappings from the set {f i | i ∈ I} of operation symbols to the set W τ (X) of all terms of type τ which preserve the arity, i.e. n i -ary operation symbols are mapped to terms which use at most the variables x 1 , . . ., x n i .
Hypersubstitutions were introduced to make precise the concept of a hyperidentity and generalizations to M -hyperidentities.
Any hypersubstitution can be uniquely extended to a map σ on W τ (X) which is inductively defined by the following steps: is an identity for every hypersubstitution σ.
Using this extension we can define a binary operation • h on the set Hyp(τ ) of all hypersubstitutions of type τ by σ 1 • h σ 2 := σ1 • σ 2 , where • is the usual composition of operations.Clearly, this makes (Hyp(τ ); • h , σ id ) a monoid with the identity σ id which maps every operation symbol f i to a so-called fundamental term f i (x 1 , . . ., x n i ).A second binary operation + can be defined on Hyp(τ ) by (σ If M is any submonoid of (Hyp(τ ), • h ), then an identity s ≈ t of a variety V is called an M -hyperidentity in V if for every σ ∈ M the equation Complexity of hypersubstitutions and lattices of varieties 35 σ[s] ≈ σ[t] is an identity in V .Here we remark that not every submonoid of (Hyp(τ ), • h ) is closed under addition.
A variety V is said to be solid if every identity is satisfied as a hyperidentity or M -solid if every identity of V is an M -hyperidentity in V .The set of all solid varieties of type τ forms a complete sublattice of the lattice of all varieties of type τ and if M 1 ⊆ M 2 , then for the lattices S M 1 (τ ), S M 2 (τ ) of M -solid varieties we have S M 2 (τ ) ⊆ S M 1 (τ ).

Complexity of hypersubstitutions
Since hypersubstitutions of type τ map operation symbols to terms, we can use the definition of a valuation of a term of type τ into an algebra IN τ to define the value of a hypersubstitution.
We denote by P re(τ ) the set of all prehypersubstitutions, i.e. hypersubstitutions which do not map any operation symbol to a variable.From the definition one can easily derive some properties of the valuation of a hypersubstitution.
Proposition 2.3 .For any two hypersubstitutions σ 1 , σ 2 ∈ Hyp(τ ) and for every valuation v which satisfies the condition (OC) we have v(σ 1 + σ 2 ) ≥ v(σ 2 ) and if σ 2 ∈ P re(τ ), then v(σ 1 • h σ 2 ) ≥ v(σ 1 ).P roof.Let f i be an arbitrary operation symbol.For the operation +, we have v(( For the operation • h , since σ 2 ∈ P re(τ ), we have σ 2 does not map Here we have used the fact that σ 2 Th.Changphas and K. Denecke does not map f i to a variable, so we can write σ 2 (f i ) = f k (t 1 , . . ., t n k ) for some index k ∈ I and some terms t 1 , . . ., t n k .Then If v is a valuation of hypersubstitutions, then for every given natural number n we consider the following set of hypersubstitutions: We list some properties of H v n .
Proposition 2.4 .Let v be a valuation of hypersubstitutions of type τ which satisfies the condition (OC).Then for every n ∈ IN, H v n has the following properties: P roof.(i) Assume that n > a. Then H v n = Hyp(τ ) since hypersubstitutions which map one of the operation symbols to a single variable, have the value a. Assume that 0 ≤ n ≤ a.Let σ be an element from Hyp(τ ).For all i ∈ I, we have v(σ(f i )) ≥ a ≥ n and this means v(σ) ≥ n and σ ∈ H v n .Altogether, we have Hyp(τ ) = H v n .(ii) Assume that 0 ≤ n ≤ a.By (i) we have H v n = Hyp(τ ) which is not contained in P re(τ ).If n > a and σ ∈ H v n , then for all i ∈ I we have v(σ(f i )) ≥ n > a.Thus σ(f i ) is not a variable and σ ∈ P re(τ ).
Then we can prove: Theorem 2.5 .Let v be a valuation of hypersubstitutions of type τ which satisfies the condition (OC).Then for every n ∈ IN the set H v n forms a sub-left-seminearring of (Hyp(τ ); • h , +).P roof.Let a be the base value of v.If 0 ≤ n ≤ a, then by Proposition 2.4 (i), H v n = Hyp(τ ).If n > a, by Proposition 2.4 (ii), H v n ⊆ P re(τ ).By Proposition 2.3, H v n forms a sub-left-seminearring of (Hyp(τ ); • h , +).

Complexity of hypersubstitutions and lattices of varieties 37
Clearly, from n 1 ≤ n 2 we get H v n 1 ⊇ H v n 2 and therefore we have a chain To each left-seminearring we form the reduct to • h , add the identity hypersubstitution σ id and consider the H v n -solid varieties.Then we get a chain of complete lattices:

Applications to varieties of semigroups
As a special valuation we consider the operation symbol count op defined in the introduction.The valuation op satisfies the condition (OC) and therefore for every n ∈ IN the set H op n forms a sub-left-seminearring of the leftseminearring (Hyp(τ ); • h , +).We add the identity hypersubstitution and denote by H op n , for short, the left-seminearring (H op n ; • h , +, σ id ).Assume now that the type is τ = (2).Clearly, H op 1 is the set P re(2) of all pre-hypersubstitutions.We determine the greatest H op 2 -solid variety of semigroups.We denote by σ t for a term t ∈ W τ ({x, y}) the hypersubstitution which maps the binary operation symbol to the term t.
2 -solid variety of semigroups.P roof.We denote by f the binary operation symbol.The following hypersubstitutions belong to H op 2 : σ f (x,f (y,x)) , σ f (f (x,x),y)) , σ f (x,f (y,y)) and so does the identity hypersubstitution σ id = σ f (x,y) , since we assume that H op 2 is a left-seminearring with identity.Applying these hypersubstitutions to the associative law we get x(yz) ≈ (xy)z, xyxzxyx ≈ xyzyx, (x 2 y) 2 z ≈ x 2 y 2 z, x(yz 2 ) 2 ≈ xy 2 z 2 and applying σ f (x,f (x,x)) which also belongs to H op 2 to the associative law we get the equation x 9 ≈ x 3 .From xyxzxyx ≈ xyzyx by identification of all variables with x one obtains x 7 ≈ x 5 and then x 3 ≈ x 5 .
The greatest H op 2 -solid variety of semigroups is the class of all semigroups which satisfy the associative law as a H op 2 -hyperidentity.We denote this class by H H op 2 M odAss.Our calculations so far show that H H op 2 M odAss ⊆ V .
To prove the converse inclusion we use a result of [3].In this paper all elements of the two-generated free algebra with respect to the variety V HR were calculated, where
Since our variety V is a subvariety of V HR , the set of all elements (classes) in F V ({x, y}) is a subset of the set of all elements (classes) of F V HR ({x, y}).It was proved in [3] that every hypersubstitution σ t , where t is one of the terms listed before and containing both variables x and y, preserves the associative identity in V HR .Since we now have a subset of this set and the identities in V HR form a subset of the identities in V , all hypersubstitutions σ t , where t ∈ F V ({x, y}) preserve the associative law in V .We have only to consider the hypersubstitutions σ t where t is a term built up only by x.Because of op(σ) ≥ 2, we have only to consider σ x 3 and σ x 4 since x 3 ≈ x 5 is an identity in V .Applying these hypersubstitutions to the associative law gives x 9 ≈ x 3 and x 16 ≈ x 4 .Both equations are consequences of x 5 ≈ x 3 and therefore satisfied in V .This shows V ⊆ H H op 2 M odAss, and therefore V is the greatest H op 2 -solid variety of semigroups.It is easy to get some conditions under which a H op 2 -solid variety of semigroups is solid.Proposition 3.2 .Let V be a non-trivial variety of semigroups.Then V is solid if and only if the following conditions hold: