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Difference between revisions of "Absorption laws"

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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
 
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}
$$x\leq y\leftrightarrow x\vee y=y\tag{*}$$
+
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 \ref{*} 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 \eqref{eq:1} apply.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , Polska Akad. Nauk  (1963) {{ZBL|0122.24311}}</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981) {{ZBL|0461.08001}}</TD></TR>
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</table>

Latest revision as of 18:42, 21 September 2017

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
How to Cite This Entry:
Absorption laws. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absorption_laws&oldid=41911
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article