Join and meet explained
of a
partially ordered set
is the
supremum (least upper bound) of
denoted
and similarly, the
meet of
is the
infimum (greatest lower bound), denoted
In general, the join and meet of a subset of a partially ordered set need not exist. Join and meet are
dual to one another with respect to order inversion.
A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.[1]
The join/meet of a subset of a totally ordered set is simply the maximal/minimal element of that subset, if such an element exists.
If a subset
of a partially ordered set
is also an (upward)
directed set, then its join (if it exists) is called a
directed join or
directed supremum. Dually, if
is a downward directed set, then its meet (if it exists) is a
directed meet or
directed infimum.
Definitions
Partial order approach
Let
be a set with a partial order
and let
An element
of
is called the
(or
or
) of
and is denoted by
if the following two conditions are satisfied:
(that is,
is a
lower bound of
).
- For any
if
then
(that is,
is greater than or equal to any other lower bound of
).
The meet need not exist, either since the pair has no lower bound at all, or since none of the lower bounds is greater than all the others. However, if there is a meet of
then it is unique, since if both
are greatest lower bounds of
then
m\leqm\primeandm\prime\leqm,
and thus
[2] If not all pairs of elements from
have a meet, then the meet can still be seen as a
partial binary operation on
If the meet does exist then it is denoted
If all pairs of elements from
have a meet, then the meet is a
binary operation on
and it is easy to see that this operation fulfills the following three conditions: For any elements
-
(commutativity),
x\wedge(y\wedgez)=(x\wedgey)\wedgez
(associativity), and-
(idempotency).
Joins are defined dually with the join of
if it exists, denoted by
An element
of
is the
(or
or
) of
in
if the following two conditions are satisfied:
(that is,
is an
upper bound of
).
- For any
if
then
(that is,
is less than or equal to any other upper bound of
).
Universal algebra approach
on a set
is a if it satisfies the three conditions
a,
b, and
c. The pair
is then a
meet-semilattice. Moreover, we then may define a
binary relation
on
A, by stating that
if and only if
In fact, this relation is a partial order on
Indeed, for any elements
since
by
c;
then
by
a; and
then
since then
x\wedgez=(x\wedgey)\wedgez=x\wedge(y\wedgez)=x\wedgey=x
by
b.
Both meets and joins equally satisfy this definition: a couple of associated meet and join operations yield partial orders which are the reverse of each other. When choosing one of these orders as the main ones, one also fixes which operation is considered a meet (the one giving the same order) and which is considered a join (the other one).
Equivalence of approaches
If
is a
partially ordered set, such that each pair of elements in
has a meet, then indeed
if and only if
since in the latter case indeed
is a lower bound of
and since
is the lower bound if and only if it is a lower bound. Thus, the partial order defined by the meet in the universal algebra approach coincides with the original partial order.
Conversely, if
is a
meet-semilattice, and the partial order
is defined as in the universal algebra approach, and
for some elements
then
is the greatest lower bound of
with respect to
since
and therefore
Similarly,
and if
is another lower bound of
then
whence
Thus, there is a meet defined by the partial order defined by the original meet, and the two meets coincide.
In other words, the two approaches yield essentially equivalent concepts, a set equipped with both a binary relation and a binary operation, such that each one of these structures determines the other, and fulfill the conditions for partial orders or meets, respectively.
Meets of general subsets
If
is a meet-semilattice, then the meet may be extended to a well-defined meet of any
non-empty finite set, by the technique described in
iterated binary operations. Alternatively, if the meet defines or is defined by a partial order, some subsets of
indeed have infima with respect to this, and it is reasonable to consider such an infimum as the meet of the subset. For non-empty finite subsets, the two approaches yield the same result, and so either may be taken as a definition of meet. In the case where subset of
has a meet, in fact
is a
complete lattice; for details, see
completeness (order theory).
Examples
is partially ordered in the usual way (by
) then joins are unions and meets are intersections; in symbols,
(where the similarity of these symbols may be used as a mnemonic for remembering that
denotes the join/supremum and
denotes the meet/infimum
[3]).
More generally, suppose that
is a
family of subsets of some set
that is partially ordered by
If
is closed under arbitrary unions and arbitrary intersections and if
belong to
then
But if
is not closed under unions then
exists in
if and only if there exists a unique
-smallest
such that
For example, if
l{F}=\{\{1\},\{2\},\{1,2,3\},\R\}
then
whereas if
l{F}=\{\{1\},\{2\},\{1,2,3\},\{0,1,2\},\R\}
then
does not exist because the sets
are the only upper bounds of
in
that could possibly be the upper bound
but
\{0,1,2\}\not\subseteq\{1,2,3\}
and
\{1,2,3\}\not\subseteq\{0,1,2\}.
If
l{F}=\{\{1\},\{2\},\{0,2,3\},\{0,1,3\}\}
then
does not exist because there is no upper bound of
in
References
. Steve Vickers (computer scientist). Topology via Logic. Cambridge Tracts in Theoretic Computer Science. 5. 0-521-36062-5. 1989. 0668.54001.
Notes and References
- Book: Grätzer . George . General Lattice Theory: Second edition . 21 November 2002 . Springer Science & Business Media . 978-3-7643-6996-5 . en . 52.
- Book: Hachtel . Gary D. . Somenzi . Fabio . Logic synthesis and verification algorithms . 1996 . Kluwer Academic Publishers . 0792397460 . 88 . registration.
- It can be immediately determined that supremums and infimums in this canonical, simple example
are
respectively. The similarity of the symbol
to
and of
to
may thus be used as a mnemonic for remembering that in the most general setting,
denotes the supremum (because a supremum is a bound from above, just like
is "above"
and
) while
denotes the infimum (because an infimum is a bound from below, just like
is "below"
and
). This can also be used to remember whether meets/joins are denoted by
or by
Intuition suggests that ""ing two sets together should produce their union
which looks similar to
so "join" must be denoted by
Similarly, two sets should "" at their intersection
which looks similar to
so "meet" must be denoted by