EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6679
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Menger Algebras of Alternating Terms
Thawhat Changphas
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002,
Thailand

Abstract. Let τn = (ni)i∈I be a particular language (type) of algebras such that ni = n for all i
in I; n is a positive integer. This paper aims to introduce n-ary alternating terms (alt-terms) of
type τn, based on the alternating group Alt(n) of degree n. We demonstrate that the set of all
n-ary alternating terms of type τn forms a Menger algebra of rank n; such algebra is denoted by
WAlt(n)

τn (Ωn). We prove that the algebra WAlt(n)
τn (Ωn) is free with respect to the variety VMenger

of Menger algebras of rank n, and it is freely generated by the set {ω(i,σ) : i ∈ I, σ ∈ Alt(n)}. We
introduce alternating hypersubstitutions of type τn and prove that the extension of an alternating
hypersubstitution of type τn acts as an endomorphism on the algebra WAlt(n)

τn (Ωn). Furthermore,
we have that the set of all alternating hypersubstitutions of type τn forms a monoid, denoted by
HypAlt(n)(τn). Finally, we establish that the set of all identities s ≈ t of a variety V of type τn,
where s and t are n-ary alternating terms of type τn, constitutes a congruence on the algebra
WAlt(n)

τn (Ωn). According to the monoid HypAlt(n)(τn), we investigate alternating hyperidentities
and alternating closed varieties.
2020 Mathematics Subject Classifications: 20M35, 20N15, 08A40, 08A60, 08A02
Key Words and Phrases: Alternating group, n-ary alternating term, Menger algebra, alternat-
ing hypersubstitution, alternating hyperidentity, alternating closed variety

1. Introduction and preliminaries

Karl Menger (cf. [1], pp. 21-24) introduced the notion of Menger algebra as a gener-
alization of the notion of semigroup; such algebras satisfy superassociative law, this is a
generalization of associative law.

Throughout, n stands for a positive integer.
For a nonempty set M , an n-ary operation (n-ary function) on M is f :Mn →M .

Definition 1. A pair (M,f) consists of a nonempty set M and an n-ary operation f on
M is a Menger algebra of rank n if

f(f(µ, ν1, . . . , νn), o1, . . . , on) = f(µ, f(ν1, o1, . . . , on), . . . , f(νn, o1, . . . , on))

for any µ, ν1, . . . , νn, o1, . . . , on ∈M .

DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6679

Email address: thacha@kku.ac.th (T. Changphas)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 2 of 15

Menger algebras of rank 1 are semigroups. Menger algebras of rank n are a significant
extension of the classical Menger algebras, providing a powerful tool for modeling and an-
alyzing multi-level and hierarchical systems. Their applications in fuzzy logic, probability,
and artificial intelligence make them relevant in both theoretical and applied mathemat-
ics. However, their increasing complexity with higher ranks necessitates a careful balance
between theoretical exploration and practical applicability. In theoretical mathematics,
Dicker demonstrated in 1963 that every Menger algebra of rank n is isomorphic to a
Menger algebra of n-ary operations defined on some set; the result is similar to Cayley’s
theorem for semigroups: any semigroup is isomorphic to a transformation semigroup.

Based on permutations and transformations (for example, full transformations, order-
preserving transformations, order-decreasing transformations), several kinds of terms and
of generalized terms (such as strongly full terms, full terms, order-preserving full terms,
order-decreasing full terms, and generalized full terms) are introduced and studied. Menger
algebras and unitary Menger algebras of particular terms have been extensively studied.
According to the account provided in the reference, Denecke [2] studied Menger algebras
and clone of terms of an arbitrary type. Since then several kinds of Menger algebras of par-
ticular terms have been investigated. Denecke and Freiberg [3] examined Menger algebras
of strongly full terms defined using permutations. Denecke and Jampachon [4] introduced
and studied Menger algebras of full terms, such terms are defined by full transformations.
By order-decreasing transformations, Wattanatripop and Changphas [5, 6] introduced and
investigated Menger algebras of order-decresing full terms. Puapong and Leeratanawalee
[7] explored Menger algebras of generalized full terms. Denecke and Hounnon [8] (also
Lekkoksung and Lekkoksung [9]) studied partial Menger algebras of linear terms and of
r-terms; some algebraic properties such as generating systems, homomorphic images and
freeness are investigated. Recently, Punigool, Phuapong and Chansuriya [10] introduced
and studied Menger algebras of terms defined based on transformations with restricted
range.

Definition 2. A type (or language) of algebras is a nonempty indexed sequence τ = (ni)i∈I
of nonnegative integers ni such that for each ni is assigned to a symbol fi. This integer is
called the arity (or rank) of fi, and fi is called an ni-ary operation (or function) symbol.

In particular, let τn = (ni)i∈I denote a type of algebras such that ni = n for all i ∈ I.
Drawing inspiration from research results mentioned above, this paper aims to intro-

duce n-ary alternating terms of type τn, based on the alternating group Alt(n) of degree
n. Firstly, we demonstrate that the set of all n-ary alternating terms of type τn forms
a Menger algebra of rank n; we denote this algebra by WAlt(n)

τn (Ωn). We prove that the
algebra WAlt(n)

τn (Ωn) is free with respect to the variety VMenger of Menger algebras of rank
n, and it is freely generated by the set {ω(i,σ) : i ∈ I, σ ∈ Alt(n)}. Secondly, we introduce
alternating hypersubstitutions of type τn and prove that the extension of an alternating
hypersubstitution of type τn acts as an endomorphism on WAlt(n)

τn (Ωn). Furthermore, we
have that the set of all alternating hypersubstitutions of type τn forms a monoid, denoted
by HypAlt(n)(τn). Finally, we establish that the set of all identities s ≈ t of a variety V



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 3 of 15

of type τn, where s and t are n-ary alternating terms of type τn, constitutes a congru-
ence on WAlt(n)

τn (Ωn). According to the monoid HypAlt(n)(τn), we investigate alternating
hyperidentities and alternating closed vareities.

2. Alternating Terms

Using Alt(n) the alternating group of degree n (see [11]), which is a subgroup of the
symmetric group Sn on the set {1, 2, . . . , n}, n-ary alternating terms (alt-terms) of type
τn are defined as follows:

Definition 3. Let Ωn = {ω1, ω2, . . . , ωn} denote a set of finite alphabet ω1, ω2, . . . , ωn,
called variables. Let (fi)i∈I be an indexed sequence of n-ary operation symbols according
to a type τn = (ni)i∈I ; this set is disjoint to the set Ωn. For any i ∈ I and σ ∈ Alt(n),

(1) fi(ωσ(1), ωσ(2), . . . , ωσ(n)) is an n-ary alt-term;

(2) If θ1, θ2, . . . , θn are n-ary alt-terms, then fi(θ1, θ2, . . . , θn) is an n-ary alt-term.

Let WAlt(n)
τn (Ωn) represent the smallest set of n-ary alt-terms of type τn which is closed

under finite application of (2).

Example 1. For the symmetric group S3 = {(1), (12), (13), (23), (123), (132)}, the alter-
nating group Alt(3) = {(1), (123), (132)}. Consider τ3 = (3) with 3-ary operation symbol g.
Then g(ω1, ω2, ω3), g(ω2, ω3, ω1), g(ω3, ω1, ω2), g(g(ω1, ω2, ω3), g(ω2, ω3, ω1), g(ω3, ω1, ω2))
are 3-ary alt-terms, where as ω1, ω2, ω3, g(ω2, ω1, ω3), g(ω3, ω2, ω1), g(ω1, ω3, ω2),
g(ω1, ω2, g(ω3, ω1, ω2)) are not 3-ary alt-terms.

Example 2. Consider the symmetric group S4. For τ4 = (4) with 4-ary operation symbol
f , f(ω1, ω2, ω3, ω4), f(ω2, ω1, ω4, ω3), f(ω3, ω4, ω1, ω2), f(ω4, ω3, ω2, ω1), f(ω2, ω3, ω1, ω4),
f(ω1, ω4, ω2, ω3), f(ω1, ω1, ω3, ω2), f(ω3, ω2, ω4, ω1), f(ω3, ω1, ω2, ω4), f(ω4, ω2, ω1, ω3),
f(ω1, ω3, ω4, ω2), f(ω2, ω4, ω3, ω1) ∈ W

Alt(4)
τ4 (Ω4). Clearly, ω1, ω2, ω3, ω4 /∈ W

Alt(4)
τ4 (Ω4);

hence f(ω1, ω2, ω3, f(ω1, ω2, ω3, ω4)), f(ω1, f(ω2, ω4, ω3, ω1), ω3, f(ω1, ω2, ω3, ω4)) /∈
W

Alt(4)
τ4 (Ω4), as well.

Example 3. Let us consider the symmetric group S4 again, but τ4 = (4, 4, 4) with 4-ary
operation symbols f, g and h. We have f(ω1, ω2, ω3, ω4), g(ω2, ω1, ω4, ω3), f(ω3, ω4, ω1, ω2),
h(ω4, ω3, ω2, ω1) are 4-ary alternating terms of type τ4; then

g(f(ω1, ω2, ω3, ω4), g(ω2, ω1, ω4, ω3), f(ω3, ω4, ω1, ω2), h(ω4, ω3, ω2, ω1))

and
h(h(ω4, ω3, ω2, ω1), g(ω2, ω1, ω4, ω3), f(ω3, ω4, ω1, ω2), f(ω1, ω2, ω3, ω4))

are 4-ary alternating terms of type τ4.



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 4 of 15

On W
Alt(n)
τn (Ωn), define

Sn : (WAlt(n)
τn (Ωn))

n+1 →WAlt(n)
τn (Ωn)

to be (n+ 1)-ary operation by:

(1) Sn(fi(ωσ(1), ωσ(2), . . . , ωσ(n)), ϑ1, . . . , ϑn) = fi(ϑσ(1), ϑσ(2), . . . , ϑσ(n));

(2) Sn(fi(θ1, . . . , θn), ϑ1, . . . , ϑn) = fi(S
n(θ1, ϑ1, . . . , ϑn), . . . , S

n(θn, ϑ1, . . . , ϑn))

for any fi, σ ∈ Alt(n), θ1, . . . , θn, ϑ1, . . . , ϑn ∈W
Alt(n)
τn (Ωn).

Theorem 1.
WAlt(n)

τn (Ωn) = (WAlt(n)
τn (Ωn), S

n)

is an algebra of type (n+ 1).

Proof. We claim that for any θ, θ1, θ2, . . . , θn ∈ W
Alt(n)
τn (Ωn), Sn(θ, θ1, θ2, . . . , θn) ∈

W
Alt(n)
τn (Ωn), which can be proved by induction on the number of occurrence of oper-

ation symbols in θ. Assume θ = fi(ωσ(1), ωσ(2), . . . , ωσ(n)) for some σ ∈ Alt(n). By
θ1, θ2, . . . , θn ∈W

Alt(n)
τn (Ωn),

Sn(θ, θ1, θ2, . . . , θn) = Sn(fi(ωσ(1), ωσ(2), . . . , ωσ(n)), θ1, θ2, . . . , θn)

= fi(θσ(1), θσ(2), . . . , θσ(n))

∈WAlt(n)
τn (Ωn).

Assume θ = fi(ϑ1, ϑ2, . . . , ϑn) and

Sn(ϑ1, θ1, θ2, . . . , θn), S
n(ϑ2, θ1, θ2, . . . , θn), . . . , S

n(ϑn, θ1, θ2, . . . , θn) ∈W
Alt(n)
τn (Ωn).

Then

Sn(θ, θ1, θ2, . . . , θn)

= Sn(fi(ϑ1, ϑ2, . . . , ϑn), θ1, θ2, . . . , θn)

= fi(S
n(ϑ1, θ1, θ2, . . . , θn), S

n(ϑ2, θ1, θ2, . . . , θn), . . . , S
n(ϑn, θ1, θ2, . . . , θn))

∈WAlt(n)
τn (Ωn).

So we have the claim.

Furthermore, we have

Theorem 2. The algebra WAlt(n)
τn (Xn) is a Menger algebra of rank n.

Proof. Let θ, θ1, . . . , θn, ϑ1, . . . , ϑn ∈W
Alt(n)
τn (Ωn). Suppose θ = fi(ωσ(1), . . . , ωσ(n)) for

some σ ∈ Alt(n). Then

Sn(Sn(θ, θ1, . . . , θn), ϑ1, . . . , ϑn)



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 5 of 15

= Sn(Sn(fi(ωσ(1), . . . , ωσ(n)), θ1, . . . , θn), ϑ1, . . . , ϑn)

= Sn(fi(θσ(1), θσ(2), . . . , θσ(n)), ϑ1, . . . , ϑn)

= fi(S
n(θσ(1), ϑ1, . . . , ϑn), . . . , S

n(θσ(n), ϑ1, . . . , ϑn))

= Sn(fi(ωσ(1), . . . , ωσ(n)), S
n(θ1, ϑ1, . . . , ϑn), . . . , S

n(θn, ϑ1, . . . , ϑn))

= Sn(θ, Sn(θ1, ϑ1, . . . , ϑn), . . . , S
n(θn, ϑ1, . . . , ϑn)).

Let θ = fi(θ
′
1, . . . , θ

′
n) be such that

Sn(Sn(θ′j , θ1, . . . , θn), ϑ1, . . . , ϑn) = Sn(θ′j , S
n(θ1, ϑ1, . . . , ϑn), . . . , S

n(θn, ϑ1, . . . , ϑn))

for any 1 ≤ j ≤ n. Then, by induction hypothesis, we have

Sn(Sn(θ, θ1, . . . , θn), ϑ1, . . . , ϑn)

= Sn(Sn(fi(θ
′
1, . . . , θ

′
n), θ1, . . . , θn), ϑ1, . . . , ϑn)

= Sn(fi(S
n(θ′1, θ1, . . . , θn), . . . , S

n(θ′n, θ1, . . . , θn)), ϑ1, . . . , ϑn)

= fi(S
n(Sn(θ′1, θ1, . . . , θn), ϑ1, . . . , ϑn), . . . , S

n(Sn(θ′n, θ1, . . . , θn), ϑ1, . . . , ϑn))

= fi(S
n(θ′1, S

n(θ1, ϑ1, . . . , ϑn), . . . , S
n(θn, ϑ1, . . . , ϑn)), . . . ,

Sn(θ′n, S
n(θ1, ϑ1, . . . , ϑn), . . . , S

n(θn, ϑ1, . . . , ϑn)))

= Sn(fi(θ
′
1, . . . , θ

′
n), S

n(θ1, ϑ1, . . . , ϑn), . . . , S
n(θn, ϑ1, . . . , ϑn))

= Sn(θ, Sn(θ1, ϑ1, . . . , ϑn), . . . , S
n(θn, ϑ1, . . . , ϑn)).

The proof is complete.

3. Freeness

Let VMenger denote the variety of all Menger algebras of rank n+ 1, and let

FVMenger
(Ξ) = (FVMenger

(Ξ), S̃n)

be the free algebra with respect to VMenger, freely generated by an indexed set of alphabet
of variables

Ξ = {ω(i,σ) : (i, σ) ∈ I ×Alt(n)}.

Theorem 3. The Menger algebra WAlt(n)
τn (Ωn) is free with respect to the variety VMenger,

freely generated by Ξ.

Proof. To prove the assertion we show that WAlt(n)
τn (Ωn) is isomorphic to the algebra

FVMenger
(Ξ). Define a mapping

Φ :WAlt(n)
τn (Ωn) → FVMenger

(Ξ)

by :



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 6 of 15

(1) Φ(fi(ωσ(1), . . . , ωσ(n))) = ω(i,σ);

(2) Φ(Sn(fi(ωσ(1), . . . , ωσ(n)), θ1, . . . , θn)) = S̃n(ω(i,σ),Φ(θ1), . . . ,Φ(θn))

for any (i, σ) ∈ I ×Alt(n) and θ1, . . . , θn ∈W
Alt(n)
τn (Ωn).

The mapping Φ is a homomorphism, i.e.,

Φ(Sn(θ, θ1, . . . , θn)) = S̃n(Φ(θ),Φ(θ1), . . . ,Φ(θn))

for all θ, θ1, . . . , θn ∈ W
Alt(n)
τn (Ωn). Suppose θ = fi(ωσ(1), . . . , ωσ(n)) for some i ∈ I and

σ ∈ Alt(n). Then

Φ(Sn(θ, θ1, . . . , θn))

= Φ(Sn(fi(ωσ(1), . . . , ωσ(n)), θ1, . . . , θn))

= S̃n(ω(i,σ),Φ(θ1), . . . ,Φ(θn))

= S̃n(Φ(fi(ωσ(1), . . . , ωσ(n))),Φ(θ1), . . . ,Φ(θn))

= S̃n(Φ(θ),Φ(θ1), . . . ,Φ(θn)).

Let θ = fi(θ
′
1, . . . , θ

′
n) such that, for 1 ≤ k ≤ n,

Φ(Sn(θ′k, θ1, . . . , θn)) = Sn(Φ(θ′k),Φ(θ1), . . . ,Φ(θn)).

From
Φ(fi(θ

′
1, . . . , θ

′
n)) = S̃n(ω(i,idn),Φ(θ

′
1), . . . ,Φ(θ

′
n))

for all θ′1, . . . , θ′n ∈W
Alt(n)
τn (Ωn) where idn is the identity map in Alt(n), we then have that

Φ(Sn(θ, θ1, . . . , θn))

= Φ(Sn(fi(θ
′
1, . . . , θ

′
n), θ1, . . . , θn))

= Φ(fi(S
n(θ′1, θ1, . . . , θn), . . . , S

n(θ′n, θ1, . . . , θn)))

= S̃n(ω(i,idn),Φ(S
n(θ′1, θ1, . . . , θn)), . . . ,Φ(S

n(θ′n, θ1, . . . , θn)))

= S̃n(ω(i,idn), S̃
n(Φ(θ′1),Φ(θ1), . . . ,Φ(θn)), . . . , S̃

n(Φ(θ′n),Φ(θ1), . . . ,Φ(θn)))

= S̃n(S̃n(y(i,idn),Φ(θ
′
1), . . . ,Φ(θ

′
n)),Φ(θ1), . . . ,Φ(θn))

= S̃n(Φ(θ),Φ(θ1), . . . ,Φ(θn)).

The mapping Φ is bijective: Assume

Φ(fi(ωσ(1), . . . , ωσ(n))) = Φ(fj(ωρ(1), . . . , ωρ(n)))

for some σ, ρ ∈ Alt(n). Then
ω(i,σ) = ω(j,ρ).

This means
(i, σ) = (j, ρ).



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 7 of 15

So
fi(ωσ(1), . . . , ωσ(n)) = fj(ωρ(1), . . . , ωρ(n)).

If ω(i,σ) ∈ Ξ, then
Φ(fi(ωσ(1), . . . , ωσ(n))) = ω(i,σ).

Hence Φ is an isomorphism, and the proof is completed.

4. Alternating Hypersubstitutions

We begin this section with introducing alternating hypersubstitutions; hypersubstitu-
tions are important notion for studying hyperidentities and solid vareities (see [12]).

Definition 4. A mapping

α : {fi : i ∈ I} →WAlt(n)
τn (Ωn)

is called an alternating hypersubstitution (or alt-hypersubstitution) of type τn.

To define the extension of an alternating hypersubstitution of type τn to the set
W

Alt(n)
τn (Ωn), for any θ ∈W

Alt(n)
τn (Ωn) and ρ ∈ Alt(n), let

(1) (θ)ρ = fi(ωρ(σ(1)), ωρ(σ(2)), . . . , ωρ(σ(n))) if θ = fi(ωσ(1), ωσ(2), . . . , ωσ(n)) for some i ∈
I, σ ∈ Alt(n);

(2) (θ)ρ = fi((θ1)ρ, (θ2)ρ, . . . , (θn)ρ) if θ = fi(θ1, θ2, . . . , θn) for some i ∈ I, θ1, θ2, . . . , θn ∈
W

Alt(n)
τn (Ωn).

It is observed that

(θ)ρ ∈W
Alt(n)
τn (Ωn)

for any ρ ∈ Alt(n) and θ ∈W
Alt(n)
τn (Ωn). Moreover,

((θ)ρ)σ = (θ)σρ

and

Sn(θ, (θ1)σ, . . . , (θn)σ)) = (Sn(θ, θ1, . . . , θn))σ

for any θ, θ1, . . . , θn ∈W
Alt(n)
τn (Ωn) and σ, ρ ∈ Alt(n).

Using notation introduced above, an alternating hypersubstitution α : {fi : i ∈ I} →
W

Alt(n)
τn (Ωn) of type τn can be extended to a mapping

α̂ :WAlt(n)
τn (Ωn) →WAlt(n)

τn (Ωn)

by:

(1) α̂[fi(ωσ(1), ωσ(2), . . . , ωσ(n))] = (α(fi))σ;



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 8 of 15

(2) α̂[fi(θ1, θ2, . . . , θn)] = Sn(α(fi), α̂[θ1], α̂[θ2], . . . , α̂[θn]).

Lemma 1. Let α be an alt-hypersubstitution of type τn. For θ, θ1, θ2, . . . , θn ∈W
Alt(n)
τn (Ωn)

and ρ ∈ Alt(n),

Sn(θ, α̂[θρ(1)], . . . , α̂[θρ(n)]) = Sn((θ)ρ, α̂[θ1], . . . , α̂[θn]).

Proof. Let θ, θ1, . . . , θn ∈W
Alt(n)
τn (Xn) and ρ ∈ Alt(n). Suppose θ = fi(ωσ(1), . . . , ωσ(n))

for some σ ∈ Alt(n). Then

Sn(θ, α̂[θρ(1)], . . . , α̂[θρ(n)]) = Sn(fi(ωσ(1), . . . , ωσ(n)), α̂[θρ(1)], . . . , α̂[θρ(n)])

= fi(α̂[θρ(σ(1))], . . . , α̂[θρ(σ(n))])

= Sn(fi(ωρ(σ(1)), . . . , ωρ(σ(n))), α̂[θ1], . . . , α̂[θn])

= Sn((θ)ρ, α̂[θ1], . . . , α̂[θn]).

Let θ = fi(θ
′
1, . . . , θ

′
n) and assume that, for 1 ≤ k ≤ n,

Sn(θ′k, α̂[θρ(1)], . . . , α̂[θρ(n)]) = Sn((θ′k)ρ, α̂[θ1], . . . , α̂[θn]).

Then

Sn(θ, α̂[θρ(1)], . . . , α̂[θρ(n)])

= Sn(fi(θ
′
1, . . . , θ

′
n), α̂[θρ(1)], . . . , α̂[θρ(n)])

= fi(S
n(θ′1, α̂[θρ(1)], . . . , α̂[θρ(n)]), . . . , S

n(θ′n, α̂[θρ(1)], . . . , α̂[θρ(n)]))

= fi(S
n((θ′1)ρ, α̂[θ1], . . . , α̂[θn]), . . . , S

n((θ′n)ρ, α̂[θ1], . . . , α̂[θn]))

= Sn(fi((θ
′
1)ρ, . . . , (θ

′
n)ρ), α̂[θ1], . . . , α̂[θn])

= Sn((θ)ρ, α̂[θ1], . . . , α̂[θn]).

Hence, the proof is complete.

Theorem 4. For any α ∈ HypAlt(n)(τn), the extension α̂ : W
Alt(n)
τn (Ωn) → W

Alt(n)
τn (Ωn)

is an endomorphism on the Menger algebra WAlt(n)
τn (Ωn).

Proof. Let α ∈ HypAlt(n)(τn). We have to show that, for any θ0, θ1, . . . , θn ∈W
Alt(n)
τn (Ωn),

α̂[Sn(θ0, θ1, . . . , θn)] = Sn(α̂[θ0], α̂[θ1], . . . , α̂[θn]).

Let θ0, θ1, . . . , θn ∈ W
Alt(n)
τn (Ωn). Suppose θ0 = fi(ωσ(1), . . . , ωσ(n)) for some i ∈ I, σ ∈

Alt(n). By Lemma 1,

α̂[Sn(θ0, θ1, . . . , θn)] = α̂[Sn(fi(ωσ(1), . . . , ωσ(n)), θ1, . . . , θn)]

= α̂[fi(θσ(1), . . . , θσ(n))]

= Sn(α(fi), α̂[θσ(1)], . . . , α̂[θσ(n)])



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 9 of 15

= Sn((α(fi))σ, α̂[θ1], . . . , α̂[θn])

= Sn(α̂[fi(ωσ(1), . . . , ωσ(n))], α̂[θ1], . . . , α̂[θn])

= Sn(α̂[θ0], α̂[θ1], . . . , α̂[θn]).

Let θ0 = fi(θ
′
1, . . . , θ

′
n), and assume, for 1 ≤ k ≤ n, that

α̂[Sn(θ′k, θ1, . . . , θn)] = Sn(α̂[θ′k], α̂[θ1], . . . , α̂[θn]).

From Theorem 3,

α̂[Sn(θ0, θ1, . . . , θn)]

= α̂[Sn(fi(θ
′
1, . . . , θ

′
n), θ1, . . . , θn)]

= α̂[fi(S
n(θ′1, θ1, . . . , θn), . . . , S

n(θ′n, θ1, . . . , θn))]

= Sn(α(fi), α̂[S
n(θ′1, θ1, . . . , θn)], . . . , α̂[S

n(θ′n, θ1, . . . , θn)])

= Sn(α(fi), S
n(α̂[θ′1], α̂[θ1], . . . , α̂[θn]), . . . , S

n(α̂[θ′n], α̂[θ1], . . . , α̂[θn]))

= Sn(Sn(α(fi), α̂[θ
′
1], . . . , α̂[θ

′
n]), α̂[θ1], . . . , α̂[θn])

= Sn(α̂[fi(θ
′
1, . . . , θ

′
n)], α̂[θ1], . . . , α̂[θn])

= Sn(α̂[θ0], α̂[θ1], . . . , α̂[θn]).

Hence
α̂[Sn(θ0, θ1, . . . , θn)] = Sn(α̂[θ0], α̂[θ1], . . . , α̂[θn]).

Define an operation ◦h on HypAlt(n)(τn) by, for any α1, α2 ∈ HypAlt(n)(τn),

(α1 ◦h α2)(fi) = α̂1[α2(fi)]

for all i ∈ I.

Lemma 2. For any α1, α2 ∈ HypAlt(n)(τn), (α1 ◦h α2)̂ = α̂1 ◦ α̂2.

Proof. At first, we prove that for any α ∈ HypAlt(n)(τn), α̂[(θ)σ] = (α̂[θ])σ for any θ ∈
W

Alt(n)
τn (Ωn) and σ ∈ Alt(n). Suppose θ = fi(ωρ(1), . . . , ωρ(n)) for some i ∈ I, ρ ∈ Alt(n).

Then (θ)σ = fi(ωσ(ρ(1)), . . . , xσ(ρ(n))); so

α̂[(θ)σ] = Sn(α(fi), ωσ(ρ(1)), . . . , ωσ(ρ(n)))

= Sn((α(fi))σρ, ω1, . . . , ωn))

= (α(fi))σρ.

Also,
(α̂[θ])σ = ((α(fi))rho)σ = (α(fi))σρ.

Let θ = fi(θ1, . . . , θn) and assume α̂[(θk)σ] = (α̂[θk])σ for any 1 ≤ k ≤ n. So

(θ)σ = fi((θ1)σ, . . . , (θn)σ).



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 10 of 15

Thus

α̂[(θ)σ] = Sn(α(fi), α̂[(θ1)σ], . . . , α̂[(θn)σ])

= Sn(α(fi), (α̂[t1])σ, . . . , (α̂[θn])σ))

= (Sn(α(fi), α̂[θ1], . . . , α̂[θn]))σ

= (α̂[θ])σ.

Now, let α1, α2 ∈ HypAlt(n)(τn), and let θ ∈ W
Alt(n)
τn (Ωn). If θ = fi(ωσ(1), . . . , ωσ(n))

for some i ∈ I, σ ∈ Alt(n), then

(α1 ◦h α2)̂[θ] = (α1 ◦h α2)̂[fi(ωσ(1), . . . , ωσ(n))] = ((α1 ◦h α2)(fi))σ = ((α̂1[α2(fi)])σ

and

(α̂1 ◦ α̂2)[θ] = (α̂1 ◦ α̂2)[fi(ωσ(1), . . . , ωσ(n))] = α̂1(α̂2[fi(ωσ(1), . . . , ωσ(n))]) = α̂1((α2(fi))σ).

Thus (α1 ◦h α2)̂[θ] = α̂1 ◦ α̂2[θ].
Let θ = fi(θ1, . . . , θn), and assume (α1 ◦h α2)̂[θk] = (α̂1 ◦ α̂2)[θk] for any 1 ≤ k ≤ n.

Then

(α1 ◦h α2)̂[θ] = (α1 ◦h α2)̂[fi(θ1, . . . , θn)]

= Sn((α1 ◦h α2)(fi), (α1 ◦h α2)̂[θ1], . . . , (α1 ◦h α2)̂[θn])

= Sn((α̂1[α2(fi)], α̂1[α̂2[θ1]], . . . , α̂1[α̂2[θn]])

= α̂1(S
n(α2(fi), α̂2[θ1], . . . , α̂2[θn]))

= α̂1[α̂2[fi(θ1, . . . , θn)]]

= (α̂1 ◦ α̂2)[θ].

Therefore the assertion holds.

The mapping αid : {fi : i ∈ I} →W
Alt(n)
τn (Ωn) which is defined by, for all i ∈ I,

αid(fi) = fi(ω1, . . . , ωn)

is an alt-hypersubstitution of type τn.

Theorem 5. HypAlt(n)(τn) = (HypAlt(n)(τn), ◦h, αid) forms a monoid.

Proof. Let α1, α2 ∈ HypAlt(n)(τn). Then, for any fi, we have (α1◦hα2)(fi) = α̂1[α2(fi)].
Since α2 ∈ HypAlt(n)(τn), α2(fi) ∈ W

Alt(n)
τn (Ωn). If α2(fi) = fi(ωσ(1), . . . , ωσ(n)) for some

σ ∈ Alt(n), then by α1(fi) ∈W
Alt(n)
τn (Ωn), we have

(α1 ◦h α2)(fi) = α̂1[α2(fi)] = (α1(fi))σ ∈W
Alt(n)
τn (Ωn).



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 11 of 15

Let α2(fi) = fi(θ1, . . . , θn) and assume α̂1[θ1], . . . , α̂1[θn] ∈W
Alt(n)
τn (Ωn). By

α̂1[fi(θ1, . . . , θn)] = Sn(α1(fi), α̂1[θ1], . . . , α̂1[θn])

and α1(fi) ∈ W
Alt(n)
τn (Ωn), it follows that α̂1[α2(fi)] ∈ W

Alt(n)
τn (Ωn). The associativity of

◦h follows by Lemma 2.

Lemma 3. The Menger algebra WAlt(n)
τn (Ωn) of rank n is generated by

F
W

Alt(n)
τn (Ωn)

= {fi(ωσ(1), . . . , ωσ(n)) : i ∈ I, σ ∈ Alt(n)}.

Proof. Clearly, fi(ωσ(1), . . . , ωσ(n)) ∈ W
Alt(n)
τn (Ωn) for any i ∈ I, σ ∈ Alt(n). Let

θ = fi(θ1, θ2, . . . , θn) ∈ W
Alt(n)
τn (Ωn) such that F

W
Alt(n)
τn (Ωn)

generates θ1, θ2, . . . , θn. We
have

Sn(fi(ω1, . . . , ωn), θ1, . . . , θn) = fi(θ1, . . . , θn) = θ.

Since the Menger algebra WAlt(n)
τn (Ωn) is generated by the set

F
W

Alt(n)
τn (Ωn)

= {fi(ωσ(1), . . . , ωσ(n)) : i ∈ I, σ ∈ Alt(n)}

so any mapping
η : F

W
Alt(n)
τn (Ωn)

→WAlt(n)
τn (Ωn)

can be uniquely extended to an endomorphism

η̄ :WAlt(n)
τn (Ωn) →WAlt(n)

τn (Ωn).

Such mappings are called alternating substitutions (alt-substitutions). The set of all alt-
substitutions will be denoted by SubstAlt(n)(τn).

For η1, η2 ∈ SubstAlt(n)(τn), define

η1 � η2 = η̄1 ◦ η2

where ◦ is the usual composition. The identity map idF
W

Alt(n)
τn (Ωn)

is an identity element
with respect to �. It turns out that

SubstAlt(n)(τn) = (SubstAlt(n)(τn),�, idF
W

Alt(n)
τn (Ωn)

)

forms a monoid. Let α ∈ HypAlt(n)(τn). Since α̂ : W
Alt(n)
τn (Ωn) → W

Alt(n)
τn (Ωn) is an

endomorphism and F
W

Alt(n)
τn (Ωn)

is a generating system of WAlt(n)
τn (Ωn), so α̂|F

W
Alt(n)
τn (Ωn)

is
an alt-substitution such that

σ̂|F
W

Alt(n)
τn (Ωn)

= α̂.



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 12 of 15

Define ψ : HypAlt(n)(τn) → SubstAlt(n)(τn) by

ψ(α) = α̂|F
W

Alt(n)
τn (Ωn)

for any α ∈ HypAlt(n)(τn). Let α1, α2 ∈ HypAlt(n)(τn). Then

ψ(α1 ◦h α2) = (α1 ◦h α2)
ˆ|F

W
Alt(n)
τn (Ωn)

= (α̂1 ◦ α̂2)|F
W

Alt(n)
τn (Ωn)

= α̂1|F
W

Alt(n)
τn (Ωn)

◦ α̂2|F
W

Alt(n)
τn (Ωn)

= ψ(α1) ◦ ψ(α2)

= ψ(α1)� ψ(α2).

We have ψ is a homomorphism. Clearly, ψ is injective. We conclude the result as follows.

Theorem 6. The monoid HypAlt(n)(τn) can be embedded into the monoid SubstAlt(n)(τn).

5. Alternating Hyperidentities

In this section let V be a variety of type τn. The set of all identities of V will be
represented by Id(V ); this is a congruence on the free algebra Fτn(Ωn) (see [13]). Let
IdAlt(n)(V ) denote the set of all identities θ ≈ ϑ of V such that θ, ϑ ∈W

Alt(n)
τn (Ωn), i.e.,

IdAlt(n)(V ) = (WAlt(n)
τn (Ωn))

2 ∩ Id(V ).

Lemma 4. IdAlt(n)(V ) is a congruence on the Menger algebra WAlt(n)
τn (Ωn).

Proof. Let θ ≈ ϑ, θ1 ≈ ϑ1, . . . , θn ≈ ϑn ∈ IdAlt(n)(V ). At first, we have

Sn(θ, θ1, . . . , θn) ≈ Sn(θ, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

To see this, suppose θ = fi(ωσ(1), . . . , ωσ(n)) for some i ∈ I, σ ∈ Alt(n). By the compati-
bility of Id(V ) with the operations f̄i of the absolutely free algebra Fτn(Ωn),

fi(θσ(1), . . . , θσ(n)) ≈ fi(ϑσ(1), . . . , ϑσ(n)) ∈ IdAlt(n)(V ).

Thus

Sn(fi(ωσ(1), . . . , ωσ(n)), θ1, . . . , θn) ≈ Sn(fi(ωσ(1), . . . , ωσ(n)), ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

Hence
Sn(θ, θ1, . . . , θn) ≈ Sn(θ, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

Let θ = fi(ν1, . . . , νn) and assume, for all 1 ≤ k ≤ n,

Sn(νk, θ1, . . . , θn) ≈ Sn(νk, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 13 of 15

Then
fi(S

n(ν1, θ1, . . . , θn), . . . , S
n(νn, θ1, . . . , θn))

≈ fi(S
n(ν1, ϑ1, . . . , ϑn), . . . , S

n(νn, ϑ1, . . . , ϑn)) ∈ IdAlt(n)(V ).

So
Sn(fi(ν1, . . . , νn), θ1, . . . , θn) ≈ Sn(fi(ν1, . . . , νn), ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

Hence
Sn(θ, θ1, . . . , θn) ≈ Sn(θ, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

Now, from
Sn(θ, ϑ1, . . . , ϑn) ≈ Sn(ϑ, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ),

it follows that

Sn(θ, θ1, . . . , θn) ≈ Sn(θ, ϑ1, . . . , ϑn) ≈ Sn(ϑ, ϑ1, . . . , ϑn) ∈ IdAlt(n)(V ).

Therefore the assertion holds.

Definition 5. V is said to be alternating closed (alt-closed) if

α̂[θ] ≈ α̂[ϑ] ∈ IdAlt(n)(V )

for all θ ≈ ϑ ∈ IdAlt(n)(V ) and for all α ∈ HypAlt(n)(τn).

Definition 6. A congruence IdAlt(n)(V ) on the Menger algebra WAlt(n)
τn (Ωn) is said to

be alternative fully invariant (alt-fully invariant) if IdAlt(n)(V ) is compatible with all
endomorphisms α̂ on WAlt(n)

τn (Ωn).

Theorem 7. If IdAlt(n)(V ) is alt-fully invariant, then V is alt-closed.

Proof. Assume IdAlt(n)(V ) is alt-fully invariant. Let θ ≈ ϑ ∈ IdAlt(n)(V ) and α ∈
HypAlt(n)(τn). By Theorem 4, α̂ is an endomorphism on WAlt(n)

τn (Ωn). Hence α̂[θ] ≈
α̂[ϑ] ∈ IdAlt(n)(τn), so V is alt-closed.

We have shown IdAlt(n)(V ) is a congruence on WAlt(n)
τn (Ωn). Consequently, quotient

algebra
WAlt(n)

τn (Ωn)/Id
Alt(n)(V )

belongs to VMenger. Note that

natIdAlt(n)(V ) :W
Alt(n)
τn (Ωn) →WAlt(n)

τn (Ωn)/Id
Alt(n)(V )

is a homomorphism with

natIdAlt(n)(V )(θ) = [θ]IdAlt(n)(V ).

Definition 7. An identity θ ≈ ϑ ∈ IdAlt(n)(V ) is an alternative hyperidentity (alt-
hyperidentity) if



T. Changphas / Eur. J. Pure Appl. Math, 18 (4) (2025), 6679 14 of 15

α̂[θ] ≈ α̂[ϑ] ∈ IdAlt(n)(V )

for all α ∈ HypAlt(n)(τn).

Theorem 8. Let θ ≈ ϑ ∈ IdAlt(n)(V ). If θ ≈ ϑ is an identity in WAlt(n)
τn (Ωn)/Id

Alt(n)(V ),
then θ ≈ ϑ is an alt-hyperidentity of V .

Proof. Assume θ ≈ ϑ is an identity inWAlt(n)
τn (Ωn)/Id

Alt(n)(V ), and let α ∈ HypAlt(n)(τn).
By

α̂ :WAlt(n)
τn (Ωn) →WAlt(n)

τn (Ωn)

is an endomorphism, we have

κ :WAlt(n)
τn (Ωn)/Id

Alt(n)(V ) →WAlt(n)
τn (Ωn)/Id

Alt(n)(V )

defined by
κ([θ]IdAlt(n)(V )) = [α̂[θ]]IdAlt(n)(V )

for all [θ]IdAlt(n)(V ) ∈W
Alt(n)
τn (Ωn)/Id

Alt(n)(V ) is an endomorphism. By assumption,

[θ]IdAlt(n)(V ) = [ϑ]IdAlt(n)(V ).

Then
κ([θ]IdAlt(n)(V )) = κ([ϑ]IdAlt(n)(V )).

Thus
[α̂[θ]]IdAlt(n)(V ) = [α̂[ϑ]]IdAlt(n)(V ).

Therefore
α̂[θ] ≈ α̂[ϑ] ∈ IdAlt(n)(V ).

Hence θ ≈ ϑ is an alt-hyperidentity of V .

Acknowledgements

The Research on ”Menger Algebras of Alternating Terms” by Khon Kaen University
has received funding support from the National Science, Research and Innovation Fund
(NSRF).

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