Describe a lattice using predicate language

Question

Let $P(x,y)$ be a binary predicate symbol.

How could we describe that $P$ is a lattice using a first-order predicate language?

I have thought of the following sentence:

$$ \forall x \forall y (P(x,y) \to (\exists z \exists t (P(x,y)\le P(z,t))∧\exists k \exists l (P(x,y) \ge P(k,l) ) $$

But I don't think it is correct. Can anyone help?

Answer 1

First, you need the three axioms saying that $P(x,y)$ is a partial order:

• Reflexivity: $\forall a \, P(a,a)$
• Antisymmetry: $\forall a \forall b \, (P(a,b) \land P(b,a) \to a = b)$
• Transitivity: $\forall a \forall b \forall c \, (P(a,b) \land P(b,c) \to P(a,c))$

Then, you have to consider the two specific axioms for lattices:

• Existence of join: $\forall a\forall b\exists c\;(P(a, c) \land P(b, c) \land \forall d\,(P(a,d) \land P(b,d) \rightarrow P(c,d)))$
• Existence of meet: $\forall a\forall b\exists c\;(P(c, a) \land P(c,b) \land \forall d \, (P(d,a) \land P(d,b) \rightarrow P(d,c)))$

Answer 2

What is a Lattice ? A Partially ordered set in which every two elements have a unique supremum and a unique infimum.

If so, WE have first to encode reflexivity, antisymmetry and transitivity of $P(x,y)$ :

$\forall x \ P(x,x) \land \forall x \forall y \ [P(x,y) \land P(y,x) \to x=y] \land \forall x \forall y \forall z \ [P(x,y) \land P(y,z) \to P(x,z)]$.

Of course, $P(x,y)$ is what we usually denote with $\le$.

The next step is to encode existence and unicity of supremum and infimum for every pair of elements.