# 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

$$x\leq y\leftrightarrow x\vee y=y\tag{*}$$

(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 (*) apply.

#### References

[1] | E. Rasiowa, R. Sikorski, "The mathematics of metamathematics" , Polska Akad. Nauk (1963) |

#### 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) |

**How to Cite This Entry:**

Absorption laws.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Absorption_laws&oldid=38909