(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let τ_n = (n_i)_i 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 \(τ_n\), based on the alternating group \(Alt(n)\) of degree \(n\). We demonstrate that the set of all \(n\)-ary alternating terms of type \(τ_n\) forms a Menger algebra of rank \(n\); such algebra is denoted by \(W^Alt(n)_τ_n(Ω_n)\). We prove that the algebra \(W^Alt(n)_τ_n(Ω_n)\) is free with respect to the variety \(V_Menger\) 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 \(τ_n\) and prove that the extension of an alternating hypersubstitution of type τ_n acts as an endomorphism on the algebra \(W^Alt(n)_τ_n(Ω_n)\). Furthermore, we have that the set of all alternating hypersubstitutions of type \(τ_n\) forms a monoid, denoted by Hyp^Alt(n)(τ_n). Finally, we establish that the set of all identities \(s ≈ t\) of a variety \(V\) of type \(τ_n\), where \(s\) and \(t\) are \(n\)-ary alternating terms of type \(τ_n\), constitutes a congruence on the algebra \(W^Alt(n)_τ_n(Ω_n)\). According to the monoid Hyp^Alt(n)(τ_n), we investigate alternating hyperidentities and alternating closed vareities.