Reductive modes
A mode is an idempotent and entropic algebra, i.e., every singleton is a subalgebra and each operation is a homomorphism. A mode is \(n\)-step left reductive if it has a binary term and satisfies the identity: \[ x_1\cdot (x_2\cdot \dots(x_n\cdot y)\dots)=x_1\cdot(x_2\cdot\dots(x_n\cdot z)\dots). \] By \(R_n\) we denote the variety of \(n\)-step left reductive modes, the variety \(R_n'\) of \(n\)-step right reductive modes is defined symmetrically. Main results: Theorem 1.6. Let \(k<n\) be integers and \(\circ\) denote the Mal'tsev product of varieties. Then \(R_n=R_k\circ R_{n-k}\). Corollary: Every variety of \(n\)-step left reductive binary modes is the \(n\)th Mal'tsev power of the variety of left-zero bands. Theorem 2.3. The join \(R_m\vee R'_n\) is independent and coincides with \(R_m\circ R'_n\). The authors describe explicitly an equational characterization of \(R_m\vee R'_n\).
variety of \(n\)-step left reductive modes
scientific article; zbMATH DE number 1354873