The typical countable algebra

TL;DR

This paper demonstrates that there is a unique countable lattice with typical properties, such as simplicity and universality, characterized as the Fraisse limit of finite lattices, and extends these ideas to other algebra classes.

Contribution

It establishes the existence and uniqueness of a typical countable lattice with key properties, using Fraisse theory, and generalizes the approach to other algebraic structures.

Findings

Existence of a unique countable lattice with typical properties.

L* is simple, locally finite, and universal for finite and countable locally finite lattices.

The approach extends to other algebra classes with Fraisse limits.

Abstract

We argue that it makes sense to talk about ``typical'' properties of lattices, and then show that there is, up to isomorphism, a unique countable lattice L* (the Fraisse limit of the class of finite lattices) that has all ``typical'' properties. Among these properties are: L* is simple and locally finite, every order preserving function can be interpolated by a lattice polynomial, and every finite lattice or countable locally finite lattice embeds into L*. The same arguments apply to other classes of algebras assuming they have a Fraisse limit and satisfy the finite embeddability property.