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Difference between revisions of "Absorption laws"

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Identities of the form
 
Identities of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104401.png" /></td> </tr></table>
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$$x\wedge(x\vee y)=x,\quad x\vee(x\wedge y)=x,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104403.png" /> are two-place operations on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104404.png" />. If these operations satisfy also the laws of commutativity and associativity, then the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104405.png" /> defined by the equivalence
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$x\leq y\leftrightarrow x\vee y=y\tag{*}$$
  
(or equivalently, by the equivalence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104407.png" />) is an order relation for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104408.png" /> is the infimum of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a0104409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044010.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044011.png" /> is the supremum. On the other hand, if the ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044012.png" /> contains an infimum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044013.png" /> and a supremum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044014.png" /> for any pair of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044016.png" />, then for the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010440/a01044018.png" /> the laws of absorption, commutativity and associativity, as well as the equivalence (*) apply.
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(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 \ref{*} apply.
  
 
====References====
 
====References====

Revision as of 19:06, 2 June 2016

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 \ref{*} 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://encyclopediaofmath.org/index.php?title=Absorption_laws&oldid=12933
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article