Absorption laws

Identities of the form

$$x\wedge(x\vee y)=x,\quad x\vee(x\wedge y)=x,$$

where $\wedge$ and $\vee$ are two-place operations on some set $L$. If these operations satisfy also the laws of commutativity and associativity, then the relation $x\leq y$ defined by the equivalence \begin{equation}\label{eq:1} x\leq y\leftrightarrow x\vee y=y \end{equation} (or equivalently, by the equivalence $x\leq y\leftrightarrow x\wedge y=x$) is an order relation for which $x\wedge y$ is the infimum of the elements $x$ and $y$, while $x\vee y$ is the supremum. On the other hand, if the ordered set $(L,\leq)$ contains an infimum $x\wedge y$ and a supremum $x\vee y$ for any pair of elements $x$ and $y$, then for the operations $\vee$ and $\wedge$ the laws of absorption, commutativity and associativity, as well as the equivalence \eqref{eq:1} apply.

References

[1]	H. Rasiowa, R. Sikorski, "The mathematics of metamathematics" , Polska Akad. Nauk (1963) Zbl 0122.24311

Comments

Instead of absorption laws one also uses the term absorptive laws, cf. [a1], Chapt. 2, Sect. 4.

References

[a1]	P.M. Cohn, "Universal algebra" , Reidel (1981) Zbl 0461.08001