User:KBlott/Definition/Lattice

Preamble
Let $(B_{\bot}^{\top}, \bigtriangledown, \bigtriangleup, -, \bot, \top)$ be a Boolean algebra. Let $(U_Z^I, \cup, \cap, -, Z, I)$ be a universe algebra. Let $L \in $$U$ be a class in the given universe. Let $\vee, \wedge: L \times L \to L$ be binary operations on $L$.

Definition
$(L, \vee, \wedge)$ is a lattice with respect to $\approx$ iff $(L, \vee, \wedge)$ is a half lattice with respect to $\approx$ and $(L, \wedge, \vee)$ is a half lattice with respect to $\approx$.