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Absorption laws

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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)


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. V.N. Grishin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Absorption_laws&oldid=12933
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098