Fraser–Horn–Hu property for ordered algebras

Abstract

Fraser and Horn, and independently Hu, studied varieties $\mathcal{V}$ of algebras satisfying the property that for every ${\mathbf{A}}_1,{\mathbf{A}}_2\in \mathcal{V},$ every congruence $\alpha \in \text{Con} ({\mathbf{A}}_1\times {\mathbf{A}}_2)$ is a product congruence, i.e., $\alpha ={\alpha }_1\times {\alpha }_2$ for some ${\alpha }_i\in \text{Con} {\mathbf{A}}_i,$ i = 1, 2. Varieties of rings with identity and congruence distributive varieties of algebras satisfy this property. It is easy to show that an algebra ${\mathbf{A}}_1\times {\mathbf{A}}_2$ has the Fraser–Horn–Hu property if and only if the map $({\alpha }_1,{\alpha }_2)\mapsto {\alpha }_1\times {\alpha }_2$ is a lattice isomorphism from $\text{Con} {\mathbf{A}}_1\times \text{Con} {\mathbf{A}}_2$ to $\text{Con} ({\mathbf{A}}_1\times {\mathbf{A}}_2).$ The property was later referred to in the literature as the Fraser–Horn–Hu property. It turns out that the Fraser–Horn–Hu property is a Mal’cev condition for varieties. In this paper, we generalize this property to varieties of ordered algebras. The classic result of Fraser, Horn and Hu follows as a special case.

Keywords:

• Fraser–Horn–Hu property
• Mal’cev conditions
• ordered algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 03C05
• 06F99
• 08B05
• 08A30