The Order of Generalized Hypersubstitutions of Type 𝜏=(2)

Abstract

The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of type 𝜏=(2) were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of type 𝜏=(2,2) were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of type 𝜏=(2). In this paper, we use similar methods to characterize idempotent generalized hypersubstitutions of type 𝜏=(2) and determine the order of each generalized hypersubstitution of this type. The main result is that the order is 1,2 or infinite.

1. Introduction

The concept of generalized hypersubstitutions was introduced by Leeratanavalee and Denecke [1]. We use it as a tool to study strong hyperidentities and use strong hyperidentities to classify varieties into collections called strong hypervarieties. Varieties which are closed under arbitrary application of generalized hypersubstitutions are called strongly solid.

A generalized hypersubstitution of type 𝜏=(𝑛_𝑖)_𝑖∈𝐼, for short, a generalized hypersubstitution is a mapping 𝜎 which maps each 𝑛_𝑖-ary operation symbol of type 𝜏 to the set 𝑊_𝜏(𝑋) of all terms of type 𝜏 built up by operation symbols from {𝑓_𝑖∣𝑖∈𝐼} where 𝑓_𝑖 is 𝑛_𝑖-ary and variables from a countably infinite alphabet of variables 𝑋∶={𝑥₁,𝑥₂,𝑥₃,…} which does not necessarily preserve the arity. We denote the set of all generalized hypersubstitutions of type 𝜏 by 𝐻𝑦𝑝_𝐺(𝜏). First, we define inductively the concept of generalized superposition of terms𝑆^𝑚∶𝑊_𝜏(𝑋)^𝑚+1→𝑊_𝜏(𝑋) by the following steps:

(i)if 𝑡=𝑥_𝑗,1≤𝑗≤𝑚, then 𝑆^𝑚(𝑥_𝑗,𝑡₁,…,𝑡_𝑚)∶=𝑡_𝑗;(ii) if 𝑡=𝑥_𝑗,𝑚<𝑗∈ℕ, then 𝑆^𝑚(𝑥_𝑗,𝑡₁,…,𝑡_𝑚)∶=𝑥_𝑗;(iii) if 𝑡=𝑓_𝑖(𝑠₁,…,𝑠_𝑛𝑖), then 𝑆^𝑚(𝑡,𝑡₁,…,𝑡_𝑚)∶=𝑓_𝑖(𝑆^𝑚(𝑠₁,𝑡₁,…,𝑡_𝑚),…,𝑆^𝑚(𝑠_𝑛𝑖,𝑡₁,…,𝑡_𝑚)). (1.1)

We extend a generalized hypersubstitution 𝜎 to a mapping 𝜎∶𝑊_𝜏(𝑋)→𝑊_𝜏(𝑋) inductively defined as follows:

(i)𝜎[𝑥]∶=𝑥∈𝑋;(ii)𝜎[𝑓_𝑖(𝑡₁,…,𝑡_𝑛𝑖)]∶=𝑆^𝑛𝑖(𝜎(𝑓_𝑖),𝜎[𝑡₁],…,𝜎[𝑡_𝑛𝑖]), for any 𝑛_𝑖-ary operation symbol 𝑓_𝑖 supposed that 𝜎[𝑡_𝑗], 1≤𝑗≤𝑛_𝑖 are already defined.

Then we define a binary operation ∘_𝐺 on 𝐻𝑦𝑝_𝐺(𝜏) by 𝜎₁∘_𝐺𝜎₂∶=𝜎₁∘𝜎₂ where ∘ denotes the usual composition of mappings and 𝜎₁,𝜎₂∈𝐻𝑦𝑝_𝐺(𝜏). Let 𝜎_id be the hypersubstitution which maps each 𝑛_𝑖-ary operation symbol 𝑓_𝑖 to the term 𝑓_𝑖(𝑥₁,…,𝑥_𝑛𝑖). We proved the following propositions.Proposition 1.1 (see [1]). For arbitrary terms 𝑡,𝑡₁,…,𝑡_𝑛∈𝑊_𝜏(𝑋) and for arbitrary generalized hypersubstitutions 𝜎,𝜎₁,𝜎₂ one has
(i)𝑆^𝑛(𝜎[𝑡],𝜎[𝑡₁],…,𝜎[𝑡_𝑛])=𝜎[𝑆^𝑛(𝑡,𝑡₁,…,𝑡_𝑛)];(ii)(𝜎₁∘𝜎₂)̂=𝜎₁∘𝜎₂.Proposition 1.2 (see [1]). 𝐻𝑦𝑝_𝐺(𝜏)=(𝐻𝑦𝑝_𝐺(𝜏);∘_𝐺,𝜎_id) is a monoid and the set of all hypersubstitutions of type 𝜏 forms a submonoid of 𝐻𝑦𝑝_𝐺(𝜏).

The order of the element 𝑎 is defined as the order of the cyclic subsemigroup ⟨𝑎⟩. The order of any hypersubstitution of type 𝜏=(2) was determined in [2].Theorem 1.3 (see [2]). Let 𝜏=(2) be a type. The order of any hypersubstitution of type 𝜏 is 1,2 or infinite.

In Section 4, we characterize the order of generalized hypersubstitutions of type 𝜏=(2).

2. Idempotent Elements in 𝐻𝑦𝑝_𝐺(2)

In this section, we consider especially the idempotent elements of 𝐻𝑦𝑝_𝐺(2). We have only one binary operation symbol, say 𝑓. The generalized hypersubstitution 𝜎 which maps 𝑓 to the term 𝑡 is denoted by 𝜎_𝑡. For any term 𝑡∈𝑊₍₂₎(𝑋), the set of all variables occurring in 𝑡 is denoted by var(𝑡). First, we will recall the definition of an idempotent element.

Definition 2.1. For any semigroup 𝑆, an element 𝑒∈𝑆 is called idempotent if 𝑒𝑒=𝑒. In general, by 𝐸(𝑆) we denote the set of all idempotent elements of 𝑆.

Proposition 2.2. An element 𝜎_𝑡∈𝐻𝑦𝑝_𝐺(2) is idempotent if and only if 𝜎_𝑡[𝑡]=𝑡.

Proof. Assume that 𝜎_𝑡 is idempotent, that is, 𝜎²_𝑡=𝜎_𝑡. Then 𝜎_𝑡[𝑡]=𝜎_𝑡[𝜎_𝑡(𝑓)]=𝜎²_𝑡(𝑓)=𝜎_𝑡(𝑓)=𝑡. (2.1)Conversely, let 𝜎_𝑡[𝑡]=𝑡. We have (𝜎_𝑡∘_𝐺𝜎_𝑡)(𝑓)=𝜎_𝑡[𝜎_𝑡(𝑓)]=𝜎_𝑡[𝑡]=𝑡=𝜎_𝑡(𝑓). Thus 𝜎²_𝑡=𝜎_𝑡.

Proposition 2.3. For every 𝑥_𝑖∈𝑋, 𝜎_𝑥𝑖 and 𝜎_id are idempotent.

Proof. Since for every 𝑥_𝑖∈𝑋, 𝜎_𝑥𝑖[𝑥_𝑖]=𝑥_𝑖. By Proposition 2.2 we have 𝜎_𝑥𝑖 is idempotent. 𝜎_id is idempotent because it is a neutral element.

Proposition 2.4. For every 𝑖,𝑗∈ℕ, the generalized hypersubstitutions 𝜎_{𝑓(𝑥1,𝑥𝑗)} and 𝜎_{𝑓(𝑥𝑖,𝑥2)} are idempotent.

Proof. Let 𝑖,𝑗∈ℕ. Then we have 𝜎_{𝑓(𝑥1,𝑥𝑗)}[𝑓(𝑥₁,𝑥_𝑗)]=𝑆²(𝜎_{𝑓(𝑥1,𝑥𝑗)}(𝑓),𝑥₁,𝑥_𝑗)=𝑆²(𝑓(𝑥₁,𝑥_𝑗),𝑥₁,𝑥_𝑗)=𝑓(𝑥₁,𝑥_𝑗𝜎),_{𝑓(𝑥𝑖,𝑥2)}[𝑓(𝑥_𝑖,𝑥₂)]=𝑆²(𝜎_{𝑓(𝑥𝑖,𝑥2)}(𝑓),𝑥_𝑖,𝑥₂)=𝑆²(𝑓(𝑥_𝑖,𝑥₂),𝑥_𝑖,𝑥₂)=𝑓(𝑥_𝑖,𝑥₂). (2.2)
Note that for any 𝑡∈𝑊₍₂₎(𝑋)⧵𝑋 and 𝑥₁,𝑥₂∉var(𝑡), 𝜎_𝑡 is idempotent. Since there is nothing to substitute in the term 𝜎_𝑡[𝑡], thus 𝜎_𝑡[𝑡]=𝑡.

Proposition 2.5. Let 𝑡∈𝑊₍₂₎(𝑋)⧵𝑋. Then the following propositions hold:
(i) if 𝑥₂∉var(𝑡), then 𝜎_{𝑓(𝑥1,𝑡)} is idempotent;(ii) if 𝑥₁∉var(𝑡), then 𝜎_{𝑓(𝑡,𝑥2)} is idempotent.

Proof. (i) Let 𝑥₂∉var(𝑡). Then 𝜎_{𝑓(𝑥1,𝑡)}[𝑓(𝑥₁,𝑡)]=𝑆²(𝜎_{𝑓(𝑥1,𝑡)}(𝑓),𝑥₁,𝜎_{𝑓(𝑥1,𝑡)}[𝑡])=𝑆²(𝑓(𝑥₁,𝑡),𝑥₁,𝜎_{𝑓(𝑥1,𝑡)}[𝑡])=𝑓(𝑥₁,𝑡). (2.3)
(ii) Let 𝑥₁∉var(𝑡). Then 𝜎_{𝑓(𝑡,𝑥2)}[𝑓(𝑡,𝑥₂)]=𝑆²(𝜎_{𝑓(𝑡,𝑥2)}(𝑓),𝜎_{𝑓(𝑡,𝑥2)}[𝑡],𝑥₂)=𝑆²(𝑓(𝑡,𝑥₂),𝜎_{𝑓(𝑡,𝑥2)}[𝑡],𝑥₂)=𝑓(𝑡,𝑥₂). (2.4)

3. Nonidempotent Elements of 𝐻𝑦𝑝_𝐺(2)

In this section, we characterize all elements of 𝐻𝑦𝑝_𝐺(2) which are not idempotent.

Proposition 3.1. If 𝑖,𝑗∈ℕ with 𝑖≠1 and 𝑗≠2, then 𝜎_{𝑓(𝑥2,𝑥𝑗)} and 𝜎_{𝑓(𝑥𝑖,𝑥1)} are not idempotent.

Proof. Let 𝑖,𝑗∈ℕ with 𝑖≠1 and 𝑗≠2. Since 𝑗≠2, 𝜎_{𝑓(𝑥2,𝑥𝑗)}[𝑓(𝑥₂,𝑥_𝑗)]=𝑆²(𝑓(𝑥₂,𝑥_𝑗),𝑥₂,𝑥_𝑗)≠𝑓(𝑥₂,𝑥_𝑗). Since 𝑖≠1, 𝜎_{𝑓(𝑥𝑖,𝑥1)}[𝑓(𝑥_𝑖,𝑥₁)]=𝑆²(𝑓(𝑥_𝑖,𝑥₁),𝑥_𝑖,𝑥₁)≠𝑓(𝑥_𝑖,𝑥₁).

Proposition 3.2. Let 𝑡∈𝑊₍₂₎(𝑋)⧵𝑋. Then the following propositions hold:
(i) if 𝑥₂∈var(𝑡), then 𝜎_{𝑓(𝑥1,𝑡)} is not idempotent;(ii) if 𝑥₁∈var(𝑡), then 𝜎_{𝑓(𝑡,𝑥2)} is not idempotent;(ii)𝜎_{𝑓(𝑡,𝑥1)} and 𝜎_{𝑓(𝑥2,𝑡)} are not idempotent;(iv) if 𝑥₁∈var(𝑡) or 𝑥₂∈var(𝑡), then 𝜎_{𝑓(𝑥𝑖,𝑡)} and 𝜎_{𝑓(𝑡,𝑥𝑖)} are not idempotent for any 𝑖∈ℕ with 𝑖>2.

Proof. (i) Let 𝑥₂∈var(𝑡). Then we have 𝜎_{𝑓(𝑥1,𝑡)}[𝑓(𝑥₁,𝑡)]=𝑆²(𝑓(𝑥₁,𝑡),𝑥₁,𝜎_{𝑓(𝑥1,𝑡)}[𝑡]). Since 𝑥₂∈var(𝑡), then we have to substitute 𝑥₂ in the term 𝑡 by 𝜎_{𝑓(𝑥1,𝑡)}[𝑡]. 𝑆²(𝑓(𝑥₁,𝑡),𝑥₁,𝜎_{𝑓(𝑥1,𝑡)}[𝑡])≠𝑓(𝑥₁,𝑡).
The proofs of (ii), (iii), and (iv) are similar to (i).

Proposition 3.3. Let 𝑡₁,𝑡₂∈𝑊₍₂₎(𝑋)⧵𝑋. If 𝑥₁∈var(𝑡₁)∪var(𝑡₂) or 𝑥₂∈var(𝑡₁)∪var(𝑡₂), then 𝜎_{𝑓(𝑡1,𝑡2)} is not idempotent.

Proof. The proof is similar to that of Proposition 3.2.

By Sections 2 and 3, we get 𝑃_𝐺(2)∪𝐸^𝐺_𝑥1∪𝐸^𝐺_𝑥2∪𝐺∪{𝜎_id} is the set of all idempotent elements in 𝐻𝑦𝑝_𝐺(2) where 𝑃_𝐺(2)∶={𝜎_𝑥𝑖∈𝐻𝑦𝑝_𝐺(2)∣𝑖∈ℕ,𝑥_𝑖∈𝑋},𝐸^𝐺_𝑥1∶={𝜎_{𝑓(𝑥1,𝑠)}∈𝐻𝑦𝑝_𝐺(2)∣𝑠∈𝑊₍₂₎(𝑋),𝑥₂∉var(𝑠)},𝐸^𝐺_𝑥2∶={𝜎_{𝑓(𝑠,𝑥2)}∈𝐻𝑦𝑝_𝐺(2)∣𝑠∈𝑊₍₂₎(𝑋),𝑥₁∉var(𝑠)} and 𝐺∶={𝜎_𝑠∈𝐻𝑦𝑝_𝐺(2)∣𝑠∈𝑊₍₂₎(𝑋)⧵𝑋,𝑥₁,𝑥₂∉var(𝑠)}.

4. The Order of Generalized Hypersubstitutions of Type 𝜏=(2)

In this section, we characterize the order of generalized hypersubstitutions of type 𝜏=(2). First, we introduce some notations. For 𝑠,𝑓(𝑐,𝑑)∈𝑊₍₂₎(𝑋),𝑥_𝑖,𝑥_𝑗∈𝑋,𝑖,𝑗∈ℕ we denote

 𝑣𝑏(𝑠)∶= the total number of variables occurring in the term 𝑠; 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)∶= the first variable (from the left) that occurs in 𝑠; 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝑠)∶= the last variable that occurs in 𝑠; 𝑊^𝐺₍₂₎({𝑥₁})∶={𝑠∈𝑊₍₂₎(𝑋)∣𝑥₁∈var(𝑠),𝑥₂∉var(𝑠)}, 𝑊^𝐺₍₂₎({𝑥₂})∶={𝑠∈𝑊₍₂₎(𝑋)∣𝑥₂∈var(𝑠),𝑥₁∉var(𝑠)}, 𝑓(𝑐,𝑑) := the term obtained from 𝑓(𝑐,𝑑) by interchanging all occurrences of the letters 𝑥₁ and 𝑥₂, that is, 𝑓(𝑐,𝑑)=𝑆²(𝑓(𝑐,𝑑),𝑥₂,𝑥₁) and 𝑓(𝑐,𝑑)=𝑆²(𝑓(𝑐,𝑑),𝑥₂,𝑥₁); 𝑓(𝑐,𝑑)^′∶= the term defined inductively by 𝑥^′_𝑖=𝑥_𝑖 and 𝑓(𝑐,𝑑)^′=𝑓(𝑑^′,𝑐^′); _𝑥𝑖𝐶[𝑓(𝑐,𝑑)]∶= the term obtained from 𝑓(𝑐,𝑑) by replacing each of the occurrences of the letter 𝑥₁ by 𝑥_𝑖, that is, _𝑥𝑖𝐶[𝑓(𝑐,𝑑)]=𝑆²(𝑓(𝑐,𝑑),𝑥_𝑖,𝑥₂); 𝐶_𝑥𝑖[𝑓(𝑐,𝑑)]∶= the term obtained from 𝑓(𝑐,𝑑) by replacing each of the occurrences of the letter 𝑥₂ by 𝑥_𝑖, that is, 𝐶_𝑥𝑖[𝑓(𝑐,𝑑)]=𝑆²(𝑓(𝑐,𝑑),𝑥₁,𝑥_𝑖); _𝑥𝑖𝐶_𝑥𝑗[𝑓(𝑐,𝑑)]∶= the term obtained from 𝑓(𝑐,𝑑) by replacing each of the occurrences of the letter 𝑥₁ by 𝑥_𝑖 and the letter 𝑥₂ by 𝑥_𝑗, that is, _𝑥𝑖𝐶_𝑥𝑗[𝑓(𝑐,𝑑)]=𝑆²(𝑓(𝑐,𝑑),𝑥_𝑖,𝑥_𝑗).

An element 𝑎 in a semigroup 𝑆 is idempotent if and only if the order of 𝑎 is 1. Then we consider only the order of generalized hypersubstitutions of type 𝜏=(2) which are not idempotent. We have the following lemmas and propositions.

Lemma 4.1. Let 𝑓(𝑐,𝑑),𝑓(𝑢,𝑣)∈𝑊₍₂₎(𝑋) and 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑣)}=𝜎_𝑤. Then 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)) unless 𝑓(𝑐,𝑑) and 𝑓(𝑢,𝑣) match one of the following 16 possibilities:
E(1)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑣)}=𝜎_{𝑓(𝑐,𝑑)} where 𝜎_{𝑓(𝑐,𝑑)}∈𝐺;E(2)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥1,𝑥1)}=𝜎_{𝐶𝑥1[𝑓(𝑐,𝑑)]};E(3)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥2)}=𝜎_{𝑥2𝐶[𝑓(𝑐,𝑑)]};E(4)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_id=𝜎_{𝑓(𝑐,𝑑)};E(5)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥1,𝑥𝑖)}=𝜎_{𝐶𝑥𝑖[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2;E(6)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥1)}=𝜎_{𝑓(𝑐,𝑑)};E(7)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥𝑖)}=𝜎_{𝑥2𝐶𝑥𝑖[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2;E(8)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥1)}=𝜎_{𝑥𝑖𝐶𝑥1[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2;E(9)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥2)}=𝜎_{𝑥𝑖𝐶[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2;E(10)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥𝑗)}=𝜎_{𝑥𝑖𝐶𝑥𝑗[𝑓(𝑐,𝑑)]} where 𝑥_𝑖,𝑥_𝑗∈𝑋, 𝑖,𝑗>2;E(11)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥1,𝑣)}=𝜎_{𝑓(𝑐,𝑑)} where 𝑣∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₁});E(12)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑣)}=𝜎_{𝑓(𝑐,𝑑)} where 𝑣∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₁});E(13)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑣)}=𝜎_{𝑥𝑖𝐶[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2, 𝑣∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₁});E(14)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑥1)}=𝜎_{𝑓(𝑐,𝑑)} where 𝑢∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₂});E(15)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑥2)}=𝜎_{𝑓(𝑐,𝑑)} where 𝑢∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₂});E(16)𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑥𝑖)}=𝜎_{𝐶𝑥𝑖[𝑓(𝑐,𝑑)]} where 𝑥_𝑖∈𝑋, 𝑖>2, 𝑢∉𝑋, 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₂}).

Proof. Assume that 𝑓(𝑐,𝑑),𝑓(𝑢,𝑣)∈𝑊₍₂₎(𝑋) and 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑣)}=𝜎_𝑤. We want to compare 𝑣𝑏(𝑤) with 𝑣𝑏(𝑓(𝑐,𝑑)). From 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑢,𝑣)}=𝜎_𝑤, it follows that 𝑤=𝑆²(𝑓(𝑐,𝑑),𝜎_{𝑓(𝑐,𝑑)}[𝑢],𝜎_{𝑓(𝑐,𝑑)}[𝑣]). If 𝜎_{𝑓(𝑐,𝑑)}∈𝐺, then 𝑤=𝑓(𝑐,𝑑) and we have E(1). Assume that 𝜎_{𝑓(𝑐,𝑑)}∉𝐺. Then 𝑥₁∈var(𝑓(𝑐,𝑑)) or 𝑥₂∈var(𝑓(𝑐,𝑑)). We will consider the following cases.
Case 1. If 𝑢,𝑣∈𝑋, then 𝜎_{𝑓(𝑐,𝑑)}[𝑢]=𝑢 and 𝜎_{𝑓(𝑐,𝑑)}[𝑣]=𝑣. This gives 9 possible subcases:
(1)𝑢=𝑣=𝑥₁: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥1,𝑥1)}=𝜎_{𝐶𝑥1[𝑓(𝑐,𝑑)]}, which is E(2);(2)𝑢=𝑣=𝑥₂: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥2)}=𝜎_{𝑥2𝐶[𝑓(𝑐,𝑑)]}, which is E(3);(3)𝑢=𝑥₁,𝑣=𝑥₂: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_id=𝜎_{𝑓(𝑐,𝑑)}, which is E(4);(4)𝑢=𝑥₁,𝑣=𝑥_𝑖,𝑖>2: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥1,𝑥𝑖)}=𝜎_{𝐶𝑥𝑖[𝑓(𝑐,𝑑)]}, which is E(5);(5)𝑢=𝑥₂,𝑣=𝑥₁: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥1)}=𝜎_{𝑓(𝑐,𝑑)}, which is E(6);(6)𝑢=𝑥₂,𝑣=𝑥_𝑖,𝑖>2: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥2,𝑥𝑖)}=𝜎_{𝑥2𝐶𝑥𝑖[𝑓(𝑐,𝑑)]}, which is E(7);(7)𝑢=𝑥_𝑖,𝑣=𝑥₁,𝑖>2: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥1)}=𝜎_{𝑥𝑖𝐶𝑥1[𝑓(𝑐,𝑑)]}, which is E(8);(8)𝑢=𝑥_𝑖,𝑣=𝑥₂,𝑖>2: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥2)}=𝜎_{𝑥𝑖𝐶[𝑓(𝑐,𝑑)]}, which is E(9);(9)𝑢=𝑥_𝑖,𝑣=𝑥_𝑗,𝑖,𝑗>2: then we have 𝜎_{𝑓(𝑐,𝑑)}∘_𝐺𝜎_{𝑓(𝑥𝑖,𝑥𝑗)}=𝜎_{𝑥𝑖𝐶𝑥𝑗[𝑓(𝑐,𝑑)]}, which is E(10).Case 2. If 𝑢=𝑥₁ and 𝑣∉𝑋, then 𝑤=𝑆²(𝑓(𝑐,𝑑),𝑥₁,𝜎_{𝑓(𝑐,𝑑)}[𝑣]). If 𝑓(𝑐,𝑑)∈𝑊^𝐺₍₂₎({𝑥₁}), then 𝑤=𝑓(𝑐,𝑑), as in E(11). Assume that 𝑥₂∈var(𝑓(𝑐,𝑑)). Since 𝑣𝑏(𝜎_{𝑓(𝑐,𝑑)}[𝑣])>1 and we have to substitute 𝑥₂ in 𝑓(𝑐,𝑑) by 𝜎_{𝑓(𝑐,𝑑)}[𝑣], we get 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 3. 𝑢=𝑥₂ and 𝑣∉𝑋. In this case we get E(12) or 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 4. 𝑢=𝑥_𝑖,𝑖>2 and 𝑣∉𝑋. In this case we get E(13) or 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 5. 𝑢∉𝑋 and 𝑣=𝑥₁. In this case we get E(14) or 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 6. 𝑢∉𝑋 and 𝑣=𝑥₂. In this case we get E(15) or 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 7. 𝑢∉𝑋 and 𝑣=𝑥_𝑖,𝑖>2. In this case we get E(16) or 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).Case 8. If 𝑢,𝑣∉𝑋, then 𝑣𝑏(𝜎_{𝑓(𝑐,𝑑)}[𝑢])>1 and 𝑣𝑏(𝜎_{𝑓(𝑐,𝑑)}[𝑣])>1. Since 𝑣𝑏(𝜎_{𝑓(𝑐,𝑑)}[𝑢])>1, 𝑣𝑏(𝜎_{𝑓(𝑐,𝑑)}[𝑣])>1 and we have to substitute 𝑥₁ in 𝑓(𝑐,𝑑) by 𝜎_{𝑓(𝑐,𝑑)}[𝑢] or 𝑥₂ in 𝑓(𝑐,𝑑) by 𝜎_{𝑓(𝑐,𝑑)}[𝑣], we get 𝑣𝑏(𝑤)>𝑣𝑏(𝑓(𝑐,𝑑)).

Lemma 4.2. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋,𝑥₁,𝑥₂∈var(𝑠),𝑡∈𝑊₍₂₎(𝑋) and 𝑥_𝑖∈𝑋 where 𝑖∈ℕ. If 𝑥_𝑖∈var(𝑡), then 𝑥_𝑖∈var(𝜎_𝑠[𝑡]).

Proof. If 𝑡∈𝑋, then 𝑡=𝑥_𝑖. So 𝜎_𝑠[𝑡]=𝑥_𝑖 and thus 𝑥_𝑖∈var(𝜎_𝑠[𝑡]). Assume that 𝑡=𝑓(𝑡₁,𝑡₂) for some 𝑡₁,𝑡₂∈𝑊₍₂₎(𝑋). Then 𝑥_𝑖∈var(𝑡₁) or 𝑥_𝑖∈var(𝑡₂). Assume that 𝑥_𝑖∈var(𝑡₁) and 𝑥_𝑖∈var(𝜎_𝑠[𝑡₁]). Consider 𝜎_𝑠[𝑡]=𝜎_𝑠[𝑓(𝑡₁,𝑡₂)]=𝑆²(𝑠,𝜎_𝑠[𝑡₁],𝜎_𝑠[𝑡₂]). Since 𝑥₁∈var(𝑠) and 𝑥_𝑖∈var(𝜎_𝑠[𝑡₁]), we get 𝑥_𝑖∈var(𝜎_𝑠[𝑡]). By the same way, we can show that if 𝑥_𝑖∈var(𝑡₂), then 𝑥_𝑖∈var(𝜎_𝑠[𝑡]).

Lemma 4.3. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋. If 𝑥₁,𝑥₂∈var(𝑠), then 𝑥₁,𝑥₂∈var(𝜎^𝑛_𝑠(𝑓)) for all 𝑛∈ℕ.

Proof. Assume that 𝑠=𝑓(𝑠₁,𝑠₂) for some 𝑠₁,𝑠₂∈𝑊₍₂₎(𝑋). For 𝑛=1, 𝜎¹_𝑠(𝑓)=𝜎_𝑠(𝑓)=𝑠. So 𝑥₁,𝑥₂∈var(𝜎¹_𝑠(𝑓)). Assume that 𝑥₁,𝑥₂∈var(𝜎^𝑛_𝑠(𝑓)). Consider 𝜎_𝑠^𝑛+1(𝑓)=(𝜎^𝑛_𝑠∘_𝐺𝜎_𝑠)(𝑓)=𝜎^𝑛_𝑠[𝜎_𝑠(𝑓)]=𝜎^𝑛_𝑠[𝑠]=𝜎^𝑛_𝑠[𝑓(𝑠₁,𝑠₂)]=𝑆²(𝜎^𝑛_𝑠(𝑓),𝜎^𝑛_𝑠[𝑠₁],𝜎^𝑛_𝑠[𝑠₂]). If 𝑥₁,𝑥₂∈var(𝑠₁), then by Lemma 4.2 we get 𝑥₁,𝑥₂∈var(𝜎^𝑛_𝑠[𝑠₁]). Since 𝑥₁∈var(𝜎^𝑛_𝑠(𝑓)) and 𝑥₁,𝑥₂∈var(𝜎^𝑛_𝑠[𝑠₁]), we get 𝑥₁,𝑥₂∈var(𝜎_𝑠^𝑛+1(𝑓)). If 𝑠₁∈𝑊^𝐺₍₂₎({𝑥₁}), then 𝑥₂∈var(𝑠₂). By Lemma 4.2, we get 𝑥₁∈var(𝜎^𝑛_𝑠[𝑠₁]) and 𝑥₂∈var(𝜎^𝑛_𝑠[𝑠₂]). Since 𝑥₁,𝑥₂∈var(𝜎^𝑛_𝑠(𝑓)), we get 𝑥₁,𝑥₂∈var(𝜎_𝑠^𝑛+1(𝑓)). If 𝑠₁∈𝑊^𝐺₍₂₎({𝑥₂}), then by the same proof as for the case 𝑠₁∈𝑊^𝐺₍₂₎({𝑥₁}) we get 𝑥₁,𝑥₂∈var(𝜎_𝑠^𝑛+1(𝑓)). If 𝑥₁,𝑥₂∉var(𝑠₁), then 𝑥₁,𝑥₂∈var(𝑠₂). By the same proof as for the case 𝑥₁,𝑥₂∈var(𝑠₁), we get 𝑥₁,𝑥₂∈var(𝜎_𝑠^𝑛+1(𝑓)).

Lemma 4.4. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋 and 𝑠∈𝑊^𝐺₍₂₎({𝑥₁}). If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥_𝑖 where 𝑥_𝑖∈𝑋,𝑖>2, then 𝑥₁,𝑥₂∉var(𝜎²_𝑠(𝑓)).

Proof. Assume that 𝑠=𝑓(𝑠₁,𝑠₂) for some 𝑠₁,𝑠₂∈𝑊₍₂₎(𝑋). Consider 𝜎²_𝑠(𝑓)=(𝜎_𝑠∘_𝐺𝜎_𝑠)(𝑓)=𝜎_𝑠[𝜎_𝑠(𝑓)]=𝜎_𝑠[𝑠]=𝜎_𝑠[𝑓(𝑠₁,𝑠₂)]=𝑆²(𝑠,𝜎_𝑠[𝑠₁],𝜎_𝑠[𝑠₂]). If 𝑠₁∈𝑋, then 𝑠₁ is the leftmost variable of 𝑠, so 𝑠₁=𝑥_𝑖. Thus 𝜎_𝑠[𝑠₁]=𝑥_𝑖. Since 𝑠∈𝑊^𝐺₍₂₎({𝑥₁}), 𝑥₁,𝑥₂∉var(𝜎_𝑠[𝑠₁]) and 𝜎²_𝑠(𝑓)=𝑆²(𝑠,𝜎_𝑠[𝑠₁],𝜎_𝑠[𝑠₂]), we get 𝑥₁,𝑥₂∉var(𝜎²_𝑠(𝑓)). Assume that 𝑠₁=𝑓(𝑠₃,𝑠₄) for some 𝑠₃,𝑠₄∈𝑊₍₂₎(𝑋). Consider 𝜎_𝑠[𝑠₁]=𝜎_𝑠[𝑓(𝑠₃,𝑠₄)]=𝑆²(𝑠,𝜎_𝑠[𝑠₃],𝜎_𝑠[𝑠₄]). If 𝑠₃∈𝑋, then 𝑠₃ is the leftmost variable of 𝑠, so 𝑠₃=𝑥_𝑖. Thus 𝜎_𝑠[𝑠₃]=𝑥_𝑖. Since 𝑠∈𝑊^𝐺₍₂₎({𝑥₁}), 𝑥₁,𝑥₂∉var(𝜎_𝑠[𝑠₃]) and 𝜎_𝑠[𝑠₁]=𝑆²(𝑠,𝜎_𝑠[𝑠₃],𝜎_𝑠[𝑠₄]), we get 𝑥₁,𝑥₂∉var(𝜎_𝑠[𝑠₁]). This implies 𝑥₁,𝑥₂∉var(𝜎²_𝑠(𝑓)). This procedure stops after finitely many steps at 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥_𝑖.

Lemma 4.5. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋. If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥₁, then 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠(𝑓))=𝑥₁ for all 𝑛∈ℕ.

Proof. Assume that 𝑠=𝑓(𝑠₁,𝑠₂) for some 𝑠₁,𝑠₂∈𝑊₍₂₎(𝑋). For 𝑛=1, 𝜎¹_𝑠(𝑓)=𝜎_𝑠(𝑓)=𝑠. So 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎¹_𝑠(𝑓))=𝑥₁. Assume that 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠(𝑓))=𝑥₁. Consider 𝜎_𝑠^𝑛+1(𝑓)=(𝜎^𝑛_𝑠∘_𝐺𝜎_𝑠)(𝑓)=𝜎^𝑛_𝑠[𝜎_𝑠(𝑓)]=𝜎^𝑛_𝑠[𝑠]=𝜎^𝑛_𝑠[𝑓(𝑠₁,𝑠₂)]=𝑆²(𝜎^𝑛_𝑠(𝑓),𝜎^𝑛_𝑠[𝑠₁],𝜎^𝑛_𝑠[𝑠₂]). If 𝑠₁∈𝑋, then 𝑠₁ is the leftmost variable of 𝑠, so 𝑠₁=𝑥₁. Thus 𝜎^𝑛_𝑠[𝑠₁]=𝑥₁. Since 𝜎_𝑠^𝑛+1(𝑓)=𝑆²(𝜎^𝑛_𝑠(𝑓),𝜎^𝑛_𝑠[𝑠₁],𝜎^𝑛_𝑠[𝑠₂]), 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠(𝑓))=𝑥₁ and 𝜎^𝑛_𝑠[𝑠₁]=𝑥₁, we get 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎_𝑠^𝑛+1(𝑓))=𝑥₁. Assume that 𝑠₁=𝑓(𝑠₃,𝑠₄) for some 𝑠₃,𝑠₄∈𝑊₍₂₎(𝑋). Consider 𝜎^𝑛_𝑠[𝑠₁]=𝜎^𝑛_𝑠[𝑓(𝑠₃,𝑠₄)]=𝑆²(𝜎^𝑛_𝑠(𝑓),𝜎^𝑛_𝑠[𝑠₃],𝜎^𝑛_𝑠[𝑠₄]). If 𝑠₃∈𝑋, then 𝑠₃ is the leftmost variable of 𝑠, so 𝑠₃=𝑥₁. Thus 𝜎^𝑛_𝑠[𝑠₃]=𝑥₁. Since 𝜎^𝑛_𝑠[𝑠₁]=𝑆²(𝜎^𝑛_𝑠(𝑓),𝜎^𝑛_𝑠[𝑠₃],𝜎^𝑛_𝑠[𝑠₄]), 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠(𝑓))=𝑥₁ and 𝜎^𝑛_𝑠[𝑠₃]=𝑥₁, we get 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠[𝑠₁])=𝑥₁. This implies 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝜎_𝑠^𝑛+1(𝑓))=𝑥₁. This procedure stops after finitely many steps at 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥₁.

Lemma 4.6. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋 and 𝑠∈𝑊^𝐺₍₂₎({𝑥₂}). If 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥_𝑖 where 𝑥_𝑖∈𝑋,𝑖>2, then 𝑥₁,𝑥₂∉var(𝜎²_𝑠(𝑓)).

Proof. The proof is similar to the proof of Lemma 4.4.

Lemma 4.7. Let 𝑠∈𝑊₍₂₎(𝑋)⧵𝑋. If 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥₂, then 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝜎^𝑛_𝑠(𝑓))=𝑥₂ for all 𝑛∈ℕ.

Proof. The proof is similar to the proof of Lemma 4.5.

Note that {𝜎^𝑛_{𝑓(𝑥2,𝑥1)}∣𝑛∈ℕ}={𝜎_id,𝜎_{𝑓(𝑥2,𝑥1)}}, and that the order of 𝜎_{𝑓(𝑥2,𝑥1)} is 2.

Proposition 4.8. Let 𝑠∈𝑊₍₂₎(𝑋), 𝑥₁,𝑥₂∈var(𝑠), 𝜎_𝑠 be not idempotent and not equal to 𝜎_{𝑓(𝑥2,𝑥1)}. Then the order of 𝜎_𝑠 is infinite.

Proof. Let 𝑛∈ℕ. Let 𝜎^𝑛_𝑠(𝑓)=𝑤. By Lemma 4.3, we get 𝑥₁,𝑥₂∈var(𝑤). Then the equation 𝜎_𝑠^𝑛+1=𝜎^𝑛_𝑠∘_𝐺𝜎_𝑠 does not fit any of E(1) to E(16), so by Lemma 4.1, we have that the term for 𝜎_𝑠^𝑛+1 is longer than 𝑤. This implies the order of 𝜎_𝑠 is infinite.

Proposition 4.9. Let 𝑠∈𝑊^𝐺₍₂₎({𝑥₁}), and 𝜎_𝑠 be not idempotent. If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥₁, then the order of 𝜎_𝑠 is infinite.

Proof. Let 𝑛∈ℕ. Let 𝜎^𝑛_𝑠(𝑓)=𝑤. By Lemma 4.5, we get 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑤)=𝑥₁. Then the equation 𝜎_𝑠^𝑛+1=𝜎^𝑛_𝑠∘_𝐺𝜎_𝑠 does not fit any of E(1) to E(16), so by Lemma 4.1 we have that the term for 𝜎_𝑠^𝑛+1 is longer than 𝑤. This implies the order of 𝜎_𝑠 is infinite.

Proposition 4.10. Let 𝑠∈𝑊^𝐺₍₂₎({𝑥₁}) and 𝜎_𝑠 be not idempotent. If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥_𝑖 where 𝑥_𝑖∈𝑋, 𝑖>2, then the order of 𝜎_𝑠 is 2.

Proof. Let 𝜎²_𝑠(𝑓)=𝑤. By Lemma 4.4, we get 𝑥₁,𝑥₂∉var(𝑤). This implies 𝜎^𝑛_𝑠=𝜎²_𝑠 for all 𝑛∈ℕ where 𝑛≥2. So the order of 𝜎_𝑠 is 2.

Proposition 4.11. Let 𝑠∈𝑊^𝐺₍₂₎({𝑥₂}) and 𝜎_𝑠 be not idempotent. If 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥₂, then the order of 𝜎_𝑠 is infinite.

Proof. The proof is similar to the proof of Proposition 4.9.

Proposition 4.12. Let 𝑠∈𝑊^𝐺₍₂₎({𝑥₂}) and 𝜎_𝑠 be not idempotent. If 𝑟𝑖𝑔ℎ𝑡𝑚𝑜𝑠𝑡(𝑠)=𝑥_𝑖 where 𝑥_𝑖∈𝑋, 𝑖>2, then the order of 𝜎_𝑠 is 2.

Proof. The proof is similar to the proof of Proposition 4.10.

Now we have the main result.

Theorem 4.13. The order of any generalized hypersubstitution of type 𝜏=(2) is 1,2 or infinite.

Proof. Let 𝜎_𝑡∈𝐻𝑦𝑝_𝐺(2). If 𝜎_𝑡 is idempotent, then the order of 𝜎_𝑡 is 1. If 𝜎_𝑡 is not idempotent, then 𝑥₁∈var(𝑡) or 𝑥₂∈var(𝑡). Assume that 𝑥₁,𝑥₂∈var(𝑡). If 𝜎_𝑡=𝜎_{𝑓(𝑥2,𝑥1)}, then the order of 𝜎_𝑡 is 2. If 𝜎_𝑡≠𝜎_{𝑓(𝑥2,𝑥1)}, then by Proposition 4.8 we get the order of 𝜎_𝑡 is infinite. Assume that 𝑥₁∈var(𝑡) and 𝑥₂∉var(𝑡). If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑡)=𝑥₁, then by Proposition 4.9 we get the order of 𝜎_𝑡 is infinite. If 𝑙𝑒𝑓𝑡𝑚𝑜𝑠𝑡(𝑡)=𝑥_𝑖 where 𝑖>2, then by Proposition 4.10 we get the order of 𝜎_𝑡 is 2. By the same way we can show that if 𝑥₂∈var(𝑡) and 𝑥₁∉var(𝑡), then the order of 𝜎_𝑡 is 2 or infinite.

Acknowledgments

This research was supported by the Graduate School and the Faculty of Science of Chiang Mai University, Thailand. The authors would like to thank the referees for useful comments.