Menger Algebras of Alternating Terms

Abstract

Let $\tau_{n} = (n_i)_{i \in I}$ be a particular language (type) of algebras such that $n_i = n$ for all $i$ in $I$; $n$ is a positive integer. This paper aims to introduce \(n\)-ary alternating terms (alt-terms) of type \(\tau_n\), based on the alternating group \(Alt(n)\) of degree \(n\). We demonstrate that the set of all \(n\)-ary alternating terms of type \(\tau_n\) forms a Menger algebra of rank \(n\); such algebra is denoted by \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). We prove that the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\) is free with respect to the variety \(V_{Menger}\) of Menger algebras of rank \(n\), and it is freely generated by the set \(\{\omega_{(i,\sigma)} : i \in I, \sigma \in Alt(n)\}\). We introduce alternating hypersubstitutions of type \(\tau_n\) and prove that the extension of an alternating hypersubstitution of type $\tau_n$ acts as an endomorphism on the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). Furthermore, we have that the set of all alternating hypersubstitutions of type \(\tau_n\) forms a monoid, denoted by ${\mathcal Hyp}^{Alt(n)}(\tau_n)$. Finally, we establish that the set of all identities \(s \approx t\) of a variety \(V\) of type \(\tau_n\), where \(s\) and \(t\) are \(n\)-ary alternating terms of type \(\tau_n\), constitutes a congruence on the algebra \({\mathcal W}^{Alt(n)}_{\tau_n}(\Omega_n)\). According to the monoid ${\mathcal Hyp}^{Alt(n)}(\tau_n)$, we investigate alternating hyperidentities and alternating closed vareities.