Reducibility of n-ary semigroups: from quasitriviality towards idempotency

Abstract

Let X be a nonempty set. Denote by ${\mathcal {F}}^n_k$ the class of associative operations $F:X^n\rightarrow X$ satisfying the condition $F(x_1,\ldots ,x_n)\in \{x_1,\ldots ,x_n\}$ whenever at least k of the elements $x_1,\ldots ,x_n$ are equal to each other. The elements of ${\mathcal {F}}^n_1$ are said to be quasitrivial and those of ${\mathcal {F}}^n_n$ are said to be idempotent. We show that ${\mathcal {F}}^n_1=\cdots ={\mathcal {F}}^n_{n-2}\subseteq {\mathcal {F}}^n_{n-1}\subseteq {\mathcal {F}}^n_n$ and we give conditions on the set X for the last inclusions to be strict. The class ${\mathcal {F}}^n_1$ was recently characterized by Couceiro and Devillet (Algebra Universalis 80:19, 2019), who showed that its elements are reducible to binary associative operations. However, some elements of ${\mathcal {F}}^n_n$ are not reducible. In this paper, we characterize the class ${\mathcal {F}}^n_{n-1}{\setminus }{\mathcal {F}}^n_1$ and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides $n-1$.

Appendix A. Alternative proof of Corollary 1.7

We provide an alternative proof of Corollary 1.7 that does not use Couceiro and Devillet (2019, Corollary 2.3).

To this extent, we first prove the following general result.

Proposition A.1

Let $F \in {\mathcal {F}}^n_n$. The following assertions are equivalent.

• (i) F is reducible to an associative and idempotent operation $G:X^2 \rightarrow X$.

• (ii) $F((n-1)\varvec{\cdot }x,y) = F(x,(n-1)\varvec{\cdot }y)$ for any $x,y\in X$.

Proof

The implication (i) $\Rightarrow$ (ii) is straightforward. Now, let us show that (ii) implies (i). So, suppose that

$$\begin{aligned} F((n-1)\varvec{\cdot }x,y) ~=~ F(x,(n-1)\varvec{\cdot }y) \qquad x,y\in X, \end{aligned}$$

(7)

and consider the operation $G:X^2 \rightarrow X$ defined by $G(x,y) = F((n-1)\varvec{\cdot }x,y)$ for any $x,y\in X$. It is not difficult to see that G is associative and idempotent. Now, let $x_1,\ldots ,x_n\in X$ and let us show that $G^{n-1}(x_1,\ldots ,x_n)=F(x_1,\ldots ,x_n)$. Using repeatedly (7) and the idempotency of F we obtain

$$\begin{aligned} G^{n-1}(x_1,\ldots ,x_n)= & {} F^{n-1}((n-1)\varvec{\cdot }x_1,(n-1)\varvec{\cdot }x_2,\ldots ,(n-1)\varvec{\cdot }x_{n-1},x_n) \\= & {} F^{n-1}((2n-3)\varvec{\cdot }x_1,x_2,(n-1)\varvec{\cdot }x_3,\ldots ,(n-1)\varvec{\cdot }x_{n-1},x_n) \\= & {} \cdots \\= & {} F^{n-1}(((n-2)(n-1)+1)\varvec{\cdot }x_1,x_2,x_3,\ldots ,x_{n-1},x_n)\\= & {} F(x_1,\ldots ,x_n), \end{aligned}$$

which shows that F is reducible to G. $\square$

Remark 2

Let $\le$ be a total ordering on X. An operation $F:X^n \rightarrow X$ is said to be $\le$-preserving if $F(x_1,\ldots ,x_n) \le F(x_1',\ldots ,x_n')$ whenever $x_i\le x_i'$ for any $i\in \{1,\ldots ,n\}$. One of the main results of Kiss and Somlai (2019, Theorem 4.8) is that every $\le$-preserving operation $F\in {\mathcal {F}}^n_n$ is reducible to an associative, idempotent, and $\le$-preserving binary operation. To this extent, they first show (Kiss and Somlai 2019, Lemma 4.1) that any $\le$-preserving operation $F\in {\mathcal {F}}^n_n$ satisfies

$$\begin{aligned} F((n-1)\varvec{\cdot }x,y) ~=~ F(x,(n-1)\varvec{\cdot }y) \qquad x,y\in X. \end{aligned}$$

Thus, we conclude that Kiss and Somlai (2019, Theorem 4.8) is an immediate consequence of Kiss and Somlai (2019, Lemma 4.1) and Proposition A.1 above.

The following result is the key for the alternative proof of Corollary 1.7.

Proposition A.2

Let $F \in {\mathcal {F}}^n_{n-1}$. The following assertions are equivalent.

• (i) F is reducible to an associative and quasitrivial operation $G:X^2 \rightarrow X$.

• (ii) F is reducible to an associative and idempotent operation $G:X^2 \rightarrow X$.

• (iii) $F((n-1)\varvec{\cdot }x,y) = F(x,(n-1)\varvec{\cdot }y)$ for any $x,y\in X$.

• (iv) $|E_F|\le 1$.

Proof

The equivalence (i) $\Leftrightarrow$ (ii) and the implication (iii) $\Rightarrow$ (iv) are straightforward. Also, the equivalence (ii) $\Leftrightarrow$ (iii) follows from Proposition A.1. Now, let us show that (iv) implies (iii). So, suppose that $|E_F|\le 1$ and suppose to the contrary that there exist $x,y\in X$ with $x\ne y$ such that $F((n-1)\varvec{\cdot }x,y) \ne F(x,(n-1)\varvec{\cdot }y)$. We have two cases to consider. If $F((n-1)\varvec{\cdot }x,y)=y$ and $F(x,(n-1)\varvec{\cdot }y)=x$, then by Lemma 2.5 we have that $x,y\in E_F$, which contradicts our assumption on $E_F$. Otherwise, if $F((n-1)\varvec{\cdot }x,y)=x$ and $F(x,(n-1)\varvec{\cdot }y)=y$, then we have

$$\begin{aligned} x ~= & {} ~ F((n-1)\varvec{\cdot }x,y) ~=~ F((n-1)\varvec{\cdot }x,F(n\varvec{\cdot }y))\\ ~= & {} ~ F(F((n-1)\varvec{\cdot }x,y),(n-1)\varvec{\cdot }y) ~=~ F(x,(n-1)\varvec{\cdot }y) ~=~y, \end{aligned}$$

which contradicts the fact that $x\ne y$. $\square$

Proof of Corollary 1.7

This follows from Proposition A.2 and Dudek and Mukhin (2006, Lemma 1). $\square$