# Absorption laws

Identities of the form

where and are two-place operations on some set . If these operations satisfy also the laws of commutativity and associativity, then the relation defined by the equivalence

 (*)

(or equivalently, by the equivalence ) is an order relation for which is the infimum of the elements and , while is the supremum. On the other hand, if the ordered set contains an infimum and a supremum for any pair of elements and , then for the operations and 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)