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Revision difference : relative complement |
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| A complement of an element in a lattice is only defined when the lattice in question is bounded. In general, a lattice is not bounded and there are no complements to speak of. Nevertheless, if the sublattice of a lattice is bounded, we can speak of complements of an element \emph{relative} to that sublattice. |
A complement of an element in a lattice is only defined when the lattice in question is bounded. In general, a lattice is not bounded and there are no complements to speak of. Nevertheless, if the sublattice of a lattice is bounded, we can speak of complements of an element \emph{relative} to that sublattice. |
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Let $L$ be a lattice, $a$ an element of $L$, and $I=[b,c]$ an interval in $L$. The element\, $d\in L$\, is said to be a complement of $a$ \emph{relative} to $I$ if
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Let $L$ be a lattice, $a$ an element of $L$, and $I=[b,c]$ an interval in $L$. $d\in L$ is said to be a complement of $a$ \emph{relative} to $I$ if
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$$a\vee d=c\,\mbox{ and }\,a\wedge d=b.$$
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$$a\vee d=c\mbox{ and }a\wedge d=b.$$
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It is easy to see that if\, $a\le c$ and $b\le a$,\, so\, $a\in I$. Similarly, $d\in I$.
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It is easy to see that $a\le c$ and $b\le a$, so $a\in I$. Similarly, $d\in I$.
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An element $a\in L$ is said to be \emph{relatively complemented} if for every interval $I$ in $L$ with $a\in I$, it has a complement relative to $I$. The lattice $L$ itself is called a \emph{relatively complemented lattice} if every element of $L$ is relatively complemented.
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An element $a\in L$ is said to be \emph{relatively complemented} if for every interval $I$ in $L$ with $a\in I$, $a$ has a complement relative to $I$. $L$ itself is called a \emph{relatively complemented lattice} if every element of $L$ is relatively complemented.
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