Lattices freely generated by posets within a variety. Part II: Finitely generated varieties

TL;DR

This paper develops a method for constructing free lattices generated by posets within finitely generated varieties, including an efficient algorithm for modular lattices and size computations for specific cases.

Contribution

It introduces a new construction method for free lattices in finitely generated varieties and provides an algorithm for modular lattices.

Findings

Method for constructing subdirect products of finite lattices

Efficient algorithm for free modular lattices generated by posets

Cardinality listings for free lattices from specific posets

Abstract

This article is the second part of an essay dedicated to lattices freely generated by posets within a variety. The first part dealt with four easy varieties while this part is concerned with finitely generated varieties. Here we present a method of constructing a subdirect product L of a finite family F of finite lattices, exploiting a set of special elements of L deducted from F. This method is applied to free lattices generated by posets within finitely generated varieties, where in the case of the variety of modular lattices, we elaborate an efficient algorithm to compute the modular lattice M freely generated by a poset. For some posets of order six, the cardinality of M is listed.