Lattices of theories in languages without equality
In [\textit{K. Adaricheva} and \textit{J. B. Nation}, Int. J. Algebra Comput. 22, No. 7, Paper No. 1250065, 27 p. (2012; Zbl 1284.08018)] it was shown that every lattice of quasi-equational theories is isomorphic to the congruence lattice of some semilattice with operators. In [\textit{K. Adaricheva} and \textit{J. B. Nation}, ibid. 22, No. 7, Paper No. 1250066, 19 p. (2012; Zbl 1284.08019)] a partial converse was proved: under some additional restrictions on the semilattice and its operators, its congruence lattice can be represented as a lattice of quasi-equational theories. Here, it is shown that without these restrictions, the congruence lattice of a semilattice with operators is isomorphic to a lattice of implicational theories in a language which does not contain equality. The paper is concluded with an overview of other results and further questions. There is also a longer version of the paper available on the author's website. There are additional sections on models and satisfaction in both atomic and implicational theories.
scientific article; zbMATH DE number 6157720