Clones of full terms
In this paper the well-known connection between hyperidentities of an algebra and identities satisfied by the clone of this algebra is studied in a restricted setting, that of n-ary full hyperidentities and identities of the n-ary clone of term operations which are induced by full terms. We prov...
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irk-123456789-1565912019-06-19T01:26:10Z Clones of full terms Denecke, K. Jampachon, P. In this paper the well-known connection between hyperidentities of an algebra and identities satisfied by the clone of this algebra is studied in a restricted setting, that of n-ary full hyperidentities and identities of the n-ary clone of term operations which are induced by full terms. We prove that the n-ary full terms form an algebraic structure which is called a Menger algebra of rank n. For a variety V , the set IdF n V of all its identities built up by full n-ary terms forms a congruence relation on that Menger algebra. If IdF n V is closed under all full hypersubstitutions, then the variety V is called n−F−solid. We will give a characterization of such varieties and apply the results to 2 − F−solid varieties of commutative groupoids. 2004 Article Clones of full terms / K. Denecke, P. Jampachon // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 1–11. — Бібліогр.: 3 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 08A40, 08A60, 08A02, 20M35. http://dspace.nbuv.gov.ua/handle/123456789/156591 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper the well-known connection between
hyperidentities of an algebra and identities satisfied by the clone
of this algebra is studied in a restricted setting, that of n-ary full
hyperidentities and identities of the n-ary clone of term operations
which are induced by full terms. We prove that the n-ary full
terms form an algebraic structure which is called a Menger algebra
of rank n. For a variety V , the set IdF
n V of all its identities built
up by full n-ary terms forms a congruence relation on that Menger
algebra. If IdF
n V is closed under all full hypersubstitutions, then
the variety V is called n−F−solid. We will give a characterization
of such varieties and apply the results to 2 − F−solid varieties of
commutative groupoids. |
| format |
Article |
| author |
Denecke, K. Jampachon, P. |
| spellingShingle |
Denecke, K. Jampachon, P. Clones of full terms Algebra and Discrete Mathematics |
| author_facet |
Denecke, K. Jampachon, P. |
| author_sort |
Denecke, K. |
| title |
Clones of full terms |
| title_short |
Clones of full terms |
| title_full |
Clones of full terms |
| title_fullStr |
Clones of full terms |
| title_full_unstemmed |
Clones of full terms |
| title_sort |
clones of full terms |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156591 |
| citation_txt |
Clones of full terms / K. Denecke, P. Jampachon // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 1–11. — Бібліогр.: 3 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT deneckek clonesoffullterms AT jampachonp clonesoffullterms |
| first_indexed |
2025-07-14T08:59:14Z |
| last_indexed |
2025-07-14T08:59:14Z |
| _version_ |
1837612196555653120 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2004). pp. 1 – 11
c© Journal “Algebra and Discrete Mathematics”
Clones of full terms
Klaus Denecke, Prakit Jampachon
Communicated by V. I. Sushchansky
Abstract. In this paper the well-known connection between
hyperidentities of an algebra and identities satisfied by the clone
of this algebra is studied in a restricted setting, that of n-ary full
hyperidentities and identities of the n-ary clone of term operations
which are induced by full terms. We prove that the n-ary full
terms form an algebraic structure which is called a Menger algebra
of rank n. For a variety V , the set IdF
n
V of all its identities built
up by full n-ary terms forms a congruence relation on that Menger
algebra. If IdF
n
V is closed under all full hypersubstitutions, then
the variety V is called n−F−solid. We will give a characterization
of such varieties and apply the results to 2 − F−solid varieties of
commutative groupoids.
1. Full terms
Here we consider algebras of n-ary type, that is, all operation symbols
have the same fixed arity n. Let τn be such a fixed n-ary type with
operation symbols (fi)i∈I indexed by some set I. Let Xn = {x1, . . . , xn}
and let X = {x1, . . . , xn, . . .} be a countably infinite set of variables.
Then Wτn(Xn) is the set of all n-ary terms of type τn. Together with
n-ary operations f i defined by
f i : Wτn(Xn)n −→Wτn(Xn) with
(t1, . . . , tn) 7→ f i(t1, . . . , tn) := f(t1, . . . , tn)
Research of the second author supported by the Ministry of University Affairs,
Thailand
2000 Mathematics Subject Classification: 08A40, 08A60, 08A02, 20M35.
Key words and phrases: Clone, unitary Menger algebra of type τn, full hyper-
identity, n-F-solid variety .
2 Clones of full terms
Wτn(Xn) forms the absolutely free algebra
Fτn(Xn) := (Wτn(Xn); (f i)i∈I)
of type τn. There is another possibility to define an operation on the set
Wτn(Xn), namely by
Sn(xj , t1, . . . , tn) := tj for 1 ≤ j ≤ n and
Sn(fi(s1, . . . , sn), t1, . . . , tn) := fi(S
n(s1, t1, . . . , tn), . . . ,
Sn(sn, , t1, . . . , tn)).
We consider a subset of Wτn(Xn), the set of all full terms. Let Hn be
the set of all mappings s : {1, . . . , n} −→ {1, . . . , n}. Full terms of type
τn are inductively defined by:
Definition 1. (i) Let s ∈ Hn be an arbitrary function and let fi be an
operation symbol of type τn. Then fi(xs(1), . . . , xs(n)) is a full term
of type τn.
(ii) If t1, . . . , tn are full terms of type τn, then fi(t1, . . . , tn) is a full
term of type τn.
Let WF
τn
(Xn) be the set of all n-ary full terms of type τn. By de-
finition, the set WF
τn
(Xn) is closed under the operations f i. Therefore
(WF
τn
(Xn); (f i)i∈I) is a subalgebra of Fτn(Xn). Clearly, the restriction
of Sn to WF
τn
(Xn) is an operation on this set. Therefore we define a
superposition operation Sn on WF
τn
(Xn) by:
Definition 2. (i) Sn(fi(xs(1), . . . , xs(n)), t1, . . . , tn)
:= fi(ts(1), . . . , ts(n)),
(ii) Sn(fi(s1, . . . , sn), t1, . . . , tn)
:= fi(S
n(s1, t1, . . . , tn), . . . , Sn(sn, , t1, . . . , tn)).
Now we consider the algebra cloneF τn := (WF
τn
(Xn);Sn) of type n+1.
Then we have:
Proposition 1. The algebra cloneF τn satisfies the so-called superasso-
ciative law
(C) S̃n(X0, S̃
n(Y1, X1, . . . , Xn), . . . , S̃n(Yn, X1, . . . , Xn))
≈ S̃n(S̃n(X0, Y1, . . . , Yn), X1, . . . , Xn), where S̃n is an (n + 1)-ary
operation symbol and Xi, Yj are variables.
Proof. We give a proof by induction on the complexity of the full term
which is substituted for X0.
K. Denecke, P. Jampachon 3
Substituting for X0 a term of the form fi(xs(1), . . . , xs(n)) for a func-
tion s ∈ Hn, then
Sn(fi(xs(1), . . . , xs(n)), S
n(t1, s1, . . . , sn), . . . , Sn(tn, s1, . . . , sn))
= fi(S
n(ts(1), s1, . . . , sn), . . . , Sn(ts(n), s1, . . . , sn))
= Sn(fi(ts(1), . . . , ts(n)), s1, . . . , sn)
= Sn(Sn(fi(xs(1), . . . , xs(n)), t1, . . . , tn), s1, . . . , sn) by Definition 2.
If we substitute for X0 a term t = fi(r1, . . . , rn) and assume that (C)
is satisfied for r1, . . . , rn, then
Sn(fi(r1, . . . , rn), Sn(t1, s1, . . . , sn), . . . , Sn(tn, s1, . . . , sn))
= fi(S
n(r1, S
n(t1, s1, . . . , sn), . . . , Sn(tn, s1, . . . , sn)), . . . ,
Sn(rn, S
n(t1, s1, . . . , sn), . . . , Sn(tn, s1, . . . , sn)))
= Sn(fi(S
n(r1, t1, . . . , tn), . . . , Sn(rn, t1, . . . , tn)), s1, . . . , sn)
= Sn(Sn(fi(r1, . . . , rn), t1, . . . , tn), s1, . . . , sn).
The algebra cloneF τn is generated by the set
Fsτn := {fi(xs(1), . . . , xs(n)) | i ∈ I, s ∈ Hn}
of so-called "fundamental terms".
The algebra cloneF τn is an example of a Menger algebra of rank n.
Definition 3. ([3]) An algebra M = (M ; Sn) of type τ = (n + 1) is
called a Menger algebra of rank n if it satisfies the axiom (C).
Let VM
_
be the variety of all algebras satisfying (C) and let FVM
_
(Y )
be the free algebra with respect to VM
_
, freely generated by Y = {yj |
j ∈ J}, where Y is a new alphabet of individual variables indexed by
the index set J = {(i, s) | i ∈ I, s ∈ Hn}. The operation of FVM
_
(Y ) is
denoted by S̃n. Then we can prove:
Theorem 1. The algebra cloneF τn is free with respect to the variety VM
_
of Menger algebras of rank n, freely generated by the set Y .
Proof. We prove that cloneF τn is isomorphic to FVM
_
(Y ) under the map-
ping ϕ : WF
τn
(Xn) −→ FVM
_
(Y ), inductively defined by
(i) ϕ(fi(xs(1), . . . , xs(n))) = y(i,s),
(ii) ϕ(fi(ts(1), . . . , ts(n))) = S̃n(y(i,s), ϕ(t1), . . . , ϕ(tn))
where i ∈ I and s ∈ Hn.
The homomorphism property can be proved by induction on the com-
plexity of the term t0. If t0 = fi(xs(1), . . . , xs(n)) for some i and some
mapping s ∈ Hn, then
4 Clones of full terms
ϕ(Sn(fi(xs(1), . . . , xs(n)), t1, . . . , tn)) = ϕ(fi(ts(1), . . . , ts(n))
= S̃n(y(i,s), ϕ(t1), . . . , ϕ(tn))
= S̃n(ϕ(fi(xs(1), . . . , xs(n))),
ϕ(t1), . . . , ϕ(tn)).
Inductively, assume that t0 = fi(r1, . . . , rn) and that
ϕ(Sn(rj , t1, . . . , tn)) = S̃n(ϕ(rj), ϕ(t1), . . . , ϕ(tn))
for all 1 ≤ j ≤ n. Then
ϕ(Sn(fi(r1, . . . , rn), t1, . . . , tn))
= ϕ(fi(S
n(r1, t1, . . . , tn), Sn(r2, t1, . . . , tn), . . . ,
Sn(rn, t1, . . . , tn)))
= S̃n(y(i,id), ϕ(Sn(r1, t1, . . . , tn)), ϕ(Sn(r2, t1, . . . , tn)), . . . ,
ϕ(Sn(rn, t1, . . . , tn)))
= S̃n(y(i,id), S̃
n(ϕ(r1), ϕ(t1), . . . , ϕ(tn)), . . . ,
S̃n(ϕ(rn), ϕ(t1), . . . , ϕ(tn)))
= S̃n(S̃n(y(i,id), ϕ(r1), . . . , ϕ(rn)), ϕ(t1), . . . , ϕ(tn)))
= S̃n(ϕ(fi(r1, . . . , rn), ϕ(t1), . . . , ϕ(tn))).
This shows that ϕ is a homomorphism. The mapping ϕ is bijective
since {y(i,s) | i ∈ I, s ∈ Hn} is a free independent set. Therefore we have
y(i,s1) = y(j,s2) ⇒ (i, s1) = (j, s2) ⇒ i = j, s1 = s2
⇒ fi(xs1(1), . . . , xs1(n)) = fj(xs2(1), . . . , xs2(n)).
Thus ϕ is bijective on the generating sets of both algebras and there-
fore ϕ is an isomorphism.
In [2] strongly full terms of type τn were defined in the following way:
(i) fi(x1, . . . , xn) is strongly full for every i ∈ I,
(ii) if t1, . . . , tn are strongly full, then fi(t1, . . . , tn) is strongly full.
Let WSF
τn
(Xn) be the set of all strongly full terms of type τn. That means,
we obtain strongly full terms by Definition 1 if we allow for s only the
identity function. Since Sn is closed on WSF
τn
(Xn) we obtain an algebra
cloneSF τn ;= (WSF
τn
(Xn);Sn).
It is clear that full terms can be expressed as strongly full terms if we
change the type from τn to τ∗n. The operation symbols of the new type
τ∗n are all n-ary and indexed by a set J which has the same cardinality as
{(i, s) | i ∈ I, s ∈ Hn}. As a result we obtain that cloneF τn is isomorphic
to cloneSF τ
∗
n.
K. Denecke, P. Jampachon 5
2. Full hypersubstitutions and substitutions of
cloneF τn
For a full term t we need the full term ts arising from t if we map all
variables corresponding to a mapping s ∈ Hn. This can be defined in-
ductively by the following steps:
(i) If t = fi(xr(1), . . . , xr(n)) for i ∈ I, r ∈ Hn,
then ts = fi(xs(r(1)), . . . , xs(r(n))).
(ii) If t = fi(t1, . . . , tn), then ts = fi((t1)s, . . . , (tn)s).
It is clear that ts is a full term for any full term t and s ∈ Hn.
Hypersubstitutions are important to describe hyperidentities and solid
varieties. We restrict this concept to full hypersubstitutions.
Definition 4. A full hypersubstitution σ of type τn is a mapping
σ : {fi | i ∈ I} −→WF
τn
(Xn).
Note that hypersubstitutions can be defined for arbitrary terms. Every
full hypersubstitution σ can be extended to a mapping σ̂ defined on
WF
τn
(Xn) by the following steps:
(i) σ̂[fi(xs(1), . . . , xs(n))] := (σ(fi))s for every s ∈ Hn,
(ii) σ̂[fi(t1, . . . , tn)] := Sn(σ(fi), σ̂[t1], . . . , σ̂[tn]).
Let HypF (τn) be the set of all full hypersubstitutions of type τn. On
HypF (τn) we define a binary operation ◦h by σ1 ◦h σ2 := σ̂1 ◦ σ2 where
◦ denotes the usual composition of functions. Together with the iden-
tity hypersubstitution σid defined by σid(fi) := fi(x1, . . . , xn) one has a
monoid (HypF (τn); ◦h, σid). For more background on hypersubstitutions
see [1]. Then we have:
Proposition 2. Let σ ∈ HypF (τn). Then σ̂ is an endomorphism on the
algebra (WF
τn
(Xn);Sn).
Proof. Indeed σ̂ : (WF
τn
(Xn);Sn) → (WF
τn
(Xn);Sn) is a function from
(WF
τn
(Xn);Sn) into itself. Now we prove by induction on the complexity
of a term t0 that for any t1, . . . , tn ∈WF
τn
(Xn),
σ̂(Sn(t0, t1, . . . , tn)) = Sn(σ̂(t0), σ̂(t1), . . . , σ̂(tn)). (1)
First we consider t0 = fi(xs(1), . . . , xs(n)) where i ∈ I and s ∈ Hn.
Then
σ̂(Sn(fi(xs(1), . . . , xs(n)), t1, . . . , tn))
6 Clones of full terms
= σ̂(fi(ts(1), . . . , ts(n))
= Sn(σ(fi), σ̂(ts(1)), . . . , σ̂(ts(n)))
= Sn((σ(fi))s, σ̂(t1), . . . , σ̂(tn))
= Sn(σ̂(t0), σ̂(t1), . . . , σ̂(tn)).
Now assume that t0 = fi(r1, . . . , rn) where ri are full terms and that
(1) holds for each ri, 1 ≤ i ≤ n. Then
σ̂(Sn(fi(r1, . . . , rn), t1, . . . , tn))
= σ̂(fi(S
n(r1, t1, . . . , tn), Sn(r2, t1, . . . , tn), . . . ,
Sn(rn, t1, . . . , tn)))
= Sn(σ(fi), σ̂(Sn(r1, t1, . . . , tn)), σ̂(Sn(r2, t1, . . . , tn)), . . . ,
σ̂(Sn(rn, t1, . . . , tn)))
= Sn(σ(fi), S
n(σ̂(r1), σ̂(t1), . . . , σ̂(tn)), . . . ,
Sn(σ̂(rn), σ̂(t1), . . . , σ̂(tn)))
= Sn(Sn(σ(fi), σ̂(r1), . . . , σ̂(rn)), σ̂(t1), . . . , σ̂(tn))
= Sn(σ̂(fi(r1, . . . , rn)), σ̂(t1), . . . , σ̂(tn)).
Therefore σ̂ is an endomorphism.
We have seen that the free algebra cloneF τn is generated by the set
Fsτn = {fi(xs(1), . . . , xs(n)) | i ∈ I, s ∈ Hn}. Therefore any mapping η
from Fsτn into WF
τn
(Xn) can be uniquely extended to an endomorphism
η of cloneF τn. Such mappings are called full clone substitutions. Let
SubstFC be the set of all such full clone substitutions. Together with a
binary composition operation ⊙ defined by η1 ⊙ η2 := η1 ◦ η2 where ◦ is
the usual composition of functions and with the identity mapping idFsτn
on Fsτn we see that (SubstFC ;⊙, idFsτn
) is a monoid. Let End(cloneF τn)
be the monoid of endomorphisms on the algebra cloneF τn. Then we
examine the connection between these monoids and the monoid of full
hypersubstitutions of type τn.
Clearly the monoids End(cloneF τn) and (SubstFC ;⊙, idFsτn
) are iso-
morphic.
Proposition 3. The monoid (HypF (τn); ◦h, σid) can be embedded into
the monoid (SubstFC ;⊙, idFsτn
).
Proof. Let σ ∈ HypF (τn). Then by Proposition 2, σ̂ is an endomorphism
on the algebra cloneF τn. Since Fsτn = {fi(xs(1), . . . , xs(n)) | i ∈ I, s ∈
Hn} is a generating set of cloneF τn, the mapping σ̂/Fsτn
is a substitution
with σ̂/Fsτn
= σ̂. We define the mapping ψ : HypF (τn) −→ SubstFC
by ψ(σ) = σ̂/Fsτn
. Injectivity of ψ is clear. We will show that ψ is a
homomorphism. Let σ1, σ2 ∈ HypF (τn). Then ψ(σ1 ◦h σ2) = (σ1 ◦h
σ2)̂ /Fsτn
= (σ̂1 ◦ σ̂2)/Fsτn
= σ̂1 ◦ σ̂2/Fsτn
= σ̂1/Fsτn
◦ σ̂2/Fsτn
= ψ(σ1) ◦
ψ(σ2) = ψ(σ1) ⊙ ψ(σ2). Clearly, the mapping ψ preserves the identity
element.
K. Denecke, P. Jampachon 7
3. Full hyperidentities and identities in cloneF τn
Let V be a variety of type τn and let IdF
nV := WF
τn
(Xn)2 ∩ IdV be the
set of all identities of V consisting of n-ary full terms. Then we have
Proposition 4. IdF
nV is a congruence on cloneF τn.
Proof. We will prove that from r ≈ t, ri ≈ ti ∈ IdF
nV, i = 1, . . . , n, there
follows Sn(r, r1, . . . , rn) ≈ Sn(t, t1, . . . , tn) ∈ IdF
nV . At first we prove
by induction on the complexity of the term t ∈ WF
τn
(Xn) that for every
n ∈ N
+ from ti ≈ ri ∈ IdF
nV, i = 1, . . . , n there follows Sn(t, t1, . . . , tn) ≈
Sn(t, r1, . . . , rn) ∈ IdF
nV . Indeed, if t = fi(xs(1), . . . , xs(n)), i ∈ I, s ∈
Hn, then
Sn(fi(xs(1), . . . , xs(n)), t1, . . . , tn)
= fi(ts(1), . . . , ts(n))
≈ fi(rs(1), . . . , rs(n))
= Sn(fi(xs(1), . . . , xs(n)), r1, . . . , rn) ∈ IdF
nV
since IdV is compatible with the operation f i of the absolutely free al-
gebra Fτn(X) and by the definition of full terms. Assume now that
t = fi(l1, . . . , ln) ∈WF
τn
(Xn) and that for lj , 1 ≤ j ≤ n, we have already
Sn(lj , t1, . . . , tn) ≈ Sn(lj , r1, . . . , rn) ∈ IdF
nV.
Then
Sn(fi(l1, . . . , ln), t1, . . . , tn) = fi(S
n(l1, t1, . . . , tn), . . . ,
Sn(ln, t1, . . . , tn))
≈ fi(S
n(l1, r1, . . . , rn), . . . ,
Sn(ln, r1, . . . , rn))
= Sn(fi(l1, . . . , ln), r1, . . . , rn)
∈ IdF
nV .
Now we prove the implication
t ≈ r ∈ IdF
nV ⇒ Sn(t, r1, . . . , rn) ≈ Sn((r, r1, . . . , rn) ∈ IdF
nV.
This is a consequence of the fully invariance of IdnV as a congruence
on the absolutely free algebra Fτn(Xn) and the definition of full terms.
Assume now that t ≈ r, ti ≈ ri ∈ IdF
nV . Then Sn(t, t1, . . . , tn) ≈
Sn(r, t1, . . . , tn) ≈ Sn(r, r1, . . . , rn) ∈ IdF
nV .
Full hypersubstitutions can be used to define F -hyperidentities in a
variety V of type τn.
Definition 5. Let V be a variety of type τn and let IdF
nV be the set of all
identities of V consisting of n-ary full terms. Then s ≈ t ∈ IdF
nV is called
8 Clones of full terms
an n-F -hyperidentity in V if σ̂[s] ≈ σ̂[t] ∈ IdF
nV for every σ ∈ HypF (τn).
If every identity in IdF
nV is satisfied as an n-F -hyperidentity, the variety
V is called n-F -solid.
We will give a sufficient condition for the n-F -solidity of a variety V .
Proposition 5. If IdF
nV is a fully invariant congruence relation on
cloneF τn, then the variety V is n-F -solid.
Proof. Let s ≈ t ∈ IdF
nV and let σ ∈ HypF (τn) be a full hypersubstitu-
tion. Since by Proposition 2 the extension σ̂ of σ is an endomorphism of
cloneF τn, we have σ̂[s] ≈ σ̂[t] ∈ IdF
nV .
As we will show later, the opposite is not true.
By Proposition 4 we can form the quotient algebra
cloneFV := cloneF τn/Id
F
nV
which belongs to the variety of Menger algebras of rank n. There is
the following connection between clone identities and n-F -hyperidentities
in V .
Proposition 6. Let V be a variety of type τn and let s ≈ t ∈ IdF
nV . If
s ≈ t is an identity in cloneFV , then it is an n-F -hyperidentity in V .
Proof. Let s ≈ t ∈ IdF
nV be an identity in cloneFV and let σ
∈ HypF (τn). Then σ̂ ∈ End(cloneF τn) and σ̂/Fsτn
∈ SubstFC with
σ̂/Fsτn
= σ̂. By the natural mapping natIdF
nV we have
natIdF
nV ◦ σ̂/Fsτn
: {fi(xs(1), . . . , xs(n)) | i ∈ I, s ∈ Hn} → cloneFV
and this is a valuation mapping with
natIdF
nV ◦ σ̂/Fsτn
= natIdF
nV ◦ σ̂.
Then
s ≈ t ∈ Id(cloneFV ) ⇒ (natIdF
nV ◦ σ̂/Fsτn
)(s)
= (natIdF
nV ◦ σ̂/Fsτn
)(t)
⇒ (natIdF
nV ◦ σ̂)(s)
= (natIdF
nV ◦ σ̂)(t)
⇒ [σ̂[s]]IdF
n V = [σ̂[t]]IdF
n V
⇒ σ̂[s] ≈ σ̂[t] ∈ IdF
nV
for every σ ∈ HypF (τn).
This means, s ≈ t is satisfied as an n-F -hyperidentity in V .
Conversely, not every n-F -hyperidentity in V is an identity in cloneFV
as the following examples show.
K. Denecke, P. Jampachon 9
4. 2-F -solid varieties of type (2)
We ask for the greatest and the least 2-F -solid varieties of groupoids.
Theorem 2. The variety Vbig = Mod{x1x2 ≈ x2x1, x
2
1 ≈ x2
2} is the
greatest 2-F -solid variety of commutative groupoids and the variety Z =
Mod{x1x2 ≈ x3x4} of all zero semigroups is the least non-trivial one.
Proof. The class of all groupoids which satisfy the commutative law
as a full hyperidentity is the greatest 2-F -solid variety of commutative
groupoids. We denote this variety by HFMod{f(x1, x2) ≈ f(x2, x1)}
where f is a binary operation symbol and call it the full hypermodel-class
of the commutative law. Since this variety is 2-F -solid, it is closed under
the application of full hypersubstitutions. The application of the full hy-
persubstitution σx1
2 to the commutative identity gives x2
1 ≈ x2
2 and the
identity hypersubstitution gives the commutative law. (We notice that
instead of f(x1, x2) we write x1x2.) This shows HFMod{f(x1, x2) ≈
f(x2, x1)} ⊆ Vbig. We can prove the opposite inclusion by showing that
Vbig is 2-F -solid since HFMod{f(x1, x2) ≈ f(x2, x1)} is the greatest 2-
F -solid variety of commutative groupoids. To do so we need all full
hypersubstitutions. But we can restrict ourselves to all full hypersub-
stitutions of type (2) which are essential for Vbig. An easy observation
shows that for any variety V , if s ≈ t is an identity in V , if for a hyper-
substitution σ1 the equation σ̂1[s] ≈ σ̂1[t] is an identity in V and if σ2
is a hypersubstitution such that σ1(f) ≈ σ2(f) is an identity in V , then
also σ̂2[s] ≈ σ̂2[t] is an identity in V . For an arbitrary term t ∈WF
(2)(X2)
we define the term td inductively by f(x1, x2)
d = f(x2, x1), f(x2, x1)
d =
f(x1, x2), f(x1, x1)
d = f(x1, x1), f(x2, x2)
d = f(x2, x2) and if t has the
form t = f(t1, t2), t1, t2 ∈ WF
(2)(Xn) we set td = S2(f(x2, x1), t
d
1, t
d
2). We
show by induction on the complexity of t ∈ WF
(2)(Xn), that t ≈ td ∈
IdF
2 Vbig. For terms of complexity 1, that is, with one binary opera-
tion symbol this is clear. Assume that t = f(t1, t2) and that tdi ≈ t ∈
IdF
2 Vbig, i = 1, 2. Then td = S2(f(x2, x1)t
d
1, t
d
2) ≈ S2(f(x2, x1), t1, t2) =
f(t2, t1) ≈ f(t1, t2) = t ∈ IdF
2 Vbig. Now we prove that for every full
hypersubstitution σt we have σ̂t[f(x2, x1)] ≈ td ∈ IdF
2 Vbig. By defini-
tion, we have σ̂t[f(x2, x1)] = σt(f)s where s is the permutation (01). If t
has complexity 1, that is, if t ∈ {f(x1, x1), f(x1, x2), f(x2, x1), f(x2, x2)},
then
f(x1, x1)
d = f(x1, x1) ≈ f(x2, x2) = σf(x1,x1)(f)s ∈ IdF
2 Vbig,
f(x2, x2)
d = f(x2, x2) ≈ f(x1, x1) = σf(x2,x2)(f)s ∈ IdF
2 Vbig,
f(x1, x2)
d = f(x2, x1) = σf(x1,x2)(f)s, f(x2, x1)
d = f(x1, x2)
= σf(x2,x1)(f)s.
10 Clones of full terms
Assume that t = f(t1, t2), then σ̂t[f(x2, x1)] = f((t1)s, (t2)s) and
assume that (ti)s = tdi ∈ IdF
2 Vbig. Then f((t1)s, (t2)s) ≈ f(td1, t
d
2) ≈
f(td2, t
d
1) = σ̂t[f(x2, x1)]. For an arbitrary full hypersubstitution σt we
have σ̂t[f(x2, x1)] = td ≈ t = σ̂t[f(x1, x2)] ∈ IdF
2 Vbig and the commuta-
tive law is satisfied as a 2-F -hyperidentity in Vbig.
We denote by txi
the term arising from the full term t by replacing
every occurrence of x2 by x1 if i = 1 and every occurrence of x1 by
x2 if i = 2. We prove that for every full term t, the equations txi
≈
txj
, i, j ∈ {1, 2}, i 6= j, are identities in Vbig. Indeed if t has complexity
1, then we have f(x1, x1) ≈ f(x2, x2) ∈ IdF
2 Vbig. If t = f(t1, t2) and
assume that tixj
≈ tixk
, i, j, k ∈ {1, 2}, j 6= k. Then tx1
= f(t1x1
, t2x1
) ≈
f(t1x2
, t2x2
) = tx2
∈ IdF
2 Vbig.
Let σt be an arbitrary full hypersubstitution. Then σ̂t[f(x1, x1)] =
(σt(f))c1 = tc1 and tc1 = tx1
≈ tx2
= tc2 = (σt(f))c2 = σ̂t[f(x2, x1)] ∈
IdF
2 Vbig where ci ∈ H2 with ci(1) = ci(2) = i, for i = 1, 2. This shows
that f(x1, x1) ≈ f(x2, x2) is a 2-F -hyperidentity in Vbig. Altogether,
this shows the 2-F -solidity of Vbig and the equality HFMod{f(x1, x2) ≈
f(x2, x1)} = Vbig.
The next step is to show that Z is the least non-trivial 2-F -solid
variety Vl of commutative groupoids.
Clearly, Vl ⊆Mod(WF
(2)(X2)
2). But from
f(x1, x2) ≈ f(x1, x1) ≈ f(x2, x2) ∈ IdF
2 Mod(WF
(2)(X2)
2)
we obtain
f(x1, x2) ≈ f(x3, x4) ∈ IdMod(WF
(2)(X2)
2)
and this means Vl ⊆ Mod(WF
(2)(X2)
2) ⊆ Z. From (WF
(2)(X2))
2 ⊆ IdZ
we obtain Z = Mod(WF
(2)(X2)
2). There is only one full binary term over
Z, namely f(x1, x2). Therefore we have only to consider the identity
hypersubstitution and this shows that Z is 2-F -solid. Since Z is an atom
in the lattice of all varieties of groupoids, it is the least non-trivial 2-F -
solid variety of commutative groupoids.
The variety Vbig is 2-F -solid but if we apply the clone endomorphism
which maps the generator f(x1, x2) to the full term f(x1, x2) and the
generator f(x2, x1) to the full term f(x1, x1) to the identity f(x1, x2) ≈
f(x2, x1) ∈ IdF
2 Vbig, then we get the equation f(x1, x2) ≈ f(x1, x1)
which is not satisfied in Vbig since Vbig 6= Z. This means, IdF
2 Vbig is not
fully invariant and the opposite of Proposition 5 is not satisfied. Since
f(x1, x2) ≈ f(x2, x1), f(x1, x1) ≈ f(x2, x2) are 2-F -hyperidentities in
Vbig and using the compatibility, we have that f(f(x1, x2), f(x1, x1)) ≈
K. Denecke, P. Jampachon 11
f(f(x2, x1), f(x2, x2)) is a 2-F -hyperidentity in Vbig. If we apply the
valuation mapping which maps f(x1, x2) and f(x1, x1) to the full term
[f(x1, x2)]Id2Vbig
and both f(x2, x1) and f(x2, x2) to [f(x1, x1)]Id2Vbig
to
the equation f(f(x1, x2), f(x1, x1)) ≈ f(f(x2, x1), f(x2, x2)), then we ob-
tain the equation [f(x1, x2)]Id2Vbig
= [f(x1, x1)]Id2Vbig
and so f(x1, x2) ≈
f(x1, x1) is an identity in Vbig, which is a contradiction. This means,
the equation f(f(x1, x2), f(x1, x1)) ≈ f(f(x2, x1), f(x2, x2)) is a 2-F -
hyperidentity in Vbig but not an identity in cloneFVbig and hence the
opposite of Proposition 6 is not satisfied.
References
[1] Denecke, K., Wismath, S.L., Hyperidentities and Clones,Gordon and Breach Sci-
ence Publishers, 2000.
[2] Denecke, K., Freiberg, L., The Algebra of Strongly Full Terms, preprint 2003.
[3] Schein, B., Trochimenko, V. S., Algebras of multiplace functions, Semigroup Fo-
rum, Vol. 17, 1979, pp.1-64.
Contact information
Klaus Denecke University of Potsdam, Institute of Mathe-
matics, Am Neuen Palais, 14415 Potsdam,
Germany
E-Mail: kdenecke@rz.uni-potsdam.de
URL: www.math.uni-potsdam.de/
∼denecke.htm
Prakit Jampachon KhonKaen University, Department of Math-
ematics, KhonKaen, 40002 Thailand
E-Mail: prajam@.kku.ac.th
Received by the editors: 23.02.2004
and final form in 17.12.2004.
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