A Note on Congruences of Infinite Bounded Involution Lattices

TL;DR

This paper demonstrates that infinite involution lattices, pseudo-Kleene algebras, and antiortholattices can have any number of congruences within a wide range, independent of their ideal structure, under certain set-theoretic assumptions.

Contribution

It establishes the possible numbers of congruences for these algebraic structures, extending understanding of their congruence lattices regardless of ideal configurations.

Findings

Any number of congruences between 2 and the number of elements or subsets is possible.

The results hold regardless of the number of ideals, under certain set-theoretic assumptions.

The findings apply to involution lattices, pseudo-Kleene algebras, and antiortholattices.

Abstract

We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.