Does a Lattice with min and max operator reduce to a total order?

Question

If I take a lattice and I add the following axioms:

• $a\vee b = a$ or $a\vee b = b$
• $a\wedge b = a$ or $a\wedge b = b$

do I get a total order?

I suppose, in this case we would define $a \leq b$ to hold whenever $a\wedge b = a$.

Answer 1

Yes.

Given $a$ and $b$, by the assumptions you made, either $a\le b$ or $b\le a$ or both. In the case that both are true, obviously $a=b$.

To check transitivity, assume $a\le b$ and $b\le c$. This means, according to the definition, that $a\land b=a$ and $b\land c=b$. Then $a\land c = (a \land b)\land c = a\land (b\land c) = a\land b = a$, that is $a\le c$.