On congruence distributivity of ordered algebras with constants (Q2906337)

A lattice is called \(n\)-distributive (\(n\geq1\)) if it satisfies the so-called \(n\)-distributive identity NEWLINE\[NEWLINEx\wedge(y_0\vee y_1\vee\ldots\vee y_n)=(x\wedge(y_1\vee y_2\vee\ldots\vee y_n))\vee(x\wedge(y_0\vee y_2\vee\ldots\vee y_n))\vee\ldots\vee (x\wedge(y_0\vee y_1\vee\ldots\vee y_{n-1})).NEWLINE\]NEWLINE Note that for \(n=1\) this is the usual distributive law. The concept of \(n\)-distributivity was introduced by \textit{A.~P.~Huhn} [Colloq. Math. Soc. János Bolyai 14, 137--147 (1976; Zbl 0385.06009)] and obviously generalizes distributivity.NEWLINENEWLINELet \((A,F)\) be an algebra and \(\leq\) a partial ordering on \(A\). If the operations from \(F\) are monotone with respect to \(\leq\), then \((A,F,\leq)\) is called an ordered algebra. Homomorphisms between ordered algebras (of the same type) are monotone mappings preserving the operations, and order-congruences are defined as kernels of homomorphisms. These congruences form an algebraic lattice. NEWLINENEWLINENEWLINE Suppose that \({\mathcal K}\) is a class of ordered algebras of type \(\tau\) and \(0\) is a fixed nullary operation symbol in \(\tau\). For \(k\)-ary lattice terms \(s\) and \(t\) it is said that the lattice identity \(s=t\) holds for order-congruences of \({\mathcal K}\) at \(0\) if for every algebra \((A,F,\leq)\) of \({\mathcal K}\) and all order-congruences \(\Theta_1,\ldots,\Theta_k\) of \((A,F,\leq)\), the \(s(\Theta_1,\ldots,\Theta_k)\)-class of \(0\) is equal to the \(t(\Theta_1,\ldots,\Theta_k)\)-class of \(0\). If \(s=t\) is the \(n\)-distributive identity, \({\mathcal K}\) is called order-congruence \(n\)-distributive at \(0\). NEWLINENEWLINENEWLINE The main result of the present paper is the following theorem: NEWLINENEWLINENEWLINE Let \({\mathcal K}\) be a class of ordered algebras of type \(\tau\) with a nullary operation symbol \(0\), and suppose that \({\mathcal K}\) is closed with respect to subalgebras and direct products. Then \({\mathcal K}\) is order-congruence \(n\)-distributive at \(0\) if and only if \({\mathcal K}\) is order-congruence \(1\)-distributive at \(0\).NEWLINENEWLINEThe proof depends on several lemmas and a rather technical proposition. Furthermore, the authors show that their theorem is a generalization of a result of \textit{G.~Czédli} and \textit{A.~Lenkehegyi} [Acta Math. Hung. 41, 17--26 (1983; Zbl 0541.06013)]. Last but not least, they derive a Mal'tsev condition for order-congruence \(n\)-distributivity at \(0\) which generalizes a result of \textit{I.~Chajda} [Arch. Math., Brno 22, 121--124 (1986; Zbl 0615.08005)].