Meet (mathematics)

In mathematics, a "meet" on a set is defined either as the unique infimum (greatest lower bound) with respect to a partial order on the set, provided an infimum exists, or (abstractly) as a commutative and associative binary operation satisfying an idempotency law. In either case, the set together with the meet is a meet-semilattice. The two definitions yield equivalent results, except that in the partial order approach it may be possible directly to define meets of more general sets of elements. The most common context in which to find a meet is as one of the operations in a lattice.

Usually, the meet of $x$ and $y$ is denoted $x\; land\; y$.

The partial order approach

Let "A" be a set with a partial order $leq$, and let $x$ and $y$ be two elements in "A". An element $z$ of "A" is the meet (or greatest lower bound or infimum) of $x$ and $y$, if the following two conditions are satisfied::1. $z\; leq\; x$ and $z\; leq\; y$ (i.e., $z$ is a "lower bound of $x$ and $y$"); and:2. for any $w$ in "A", such that $w\; leq\; x$ and $w\; leq\; y$, we have $w\; leq\; z$ (i.e., $z$ is greater than any other lower bound of $x$ and $y$).If there is a meet of $x$ and $y$, then indeed it is unique, since if both $z$ and $z\text{'}$ are greatest lower bounds of $x$ and $y$, then $z\; leq\; z\text{'}\; leq\; z$, whence indeed $z\; =\; z\text{'},$. If the meet does exist, it is denoted $x\; land\; y$.Some pairs of elements in "A" may lack a meet, either since they have no lower bound at all, or since none of their lower bounds is greater than all the others. If all pairs of elements have meets, then indeed the meet is a binary operation on "A", and it is easy to see that this operation fulfils the following three conditions: For any elements $x$, $y$, and $z$ in "A",:a. $x\; land\; y\; =\; y\; land\; x$ (commutativity),:b. $x\; land\; (y\; land\; z)\; =(x\; land\; y)\; land\; z$ (associativity), and:c. $x\; land\; x\; =\; x$ (idempotency).

The universal algebra approach

By definition, a binary operation $land$ on a set "A" is a "meet", if it satisfies the three conditions a, b, and c supra. The pair ("A",$land$) then is a meet-semilattice. Moreover, we then may define a binary relation $leq$ on "A", by stating that $x\; leq\; y$ if and only if $x\; land\; y\; =\; x$. In fact, this relation is a partial order on "A". Indeed, for any elements $x$, $y$, and $z$ in "A",:$x\; leq\; x$, since $x\; land\; x\; =\; x$ by c;:if $x\; leq\; y$ and $y\; leq\; x$, then $x\; =\; x\; land\; y\; =\; y\; land\; x\; =\; y$ by a; and:if $x\; leq\; y$ and $y\; leq\; z$, then $x\; leq\; z$, since then $x\; land\; z\; =\; (x\; land\; y)\; land\; z\; =\; x\; land\; (y\; land\; z)\; =\; x\; land\; y\; =\; x$ by b.

Equivalence of approaches

If ("A",$leq$) is a partially ordered set, such that each pair of elements in "A" has a meet, then indeed $x\; land\; y\; =\; x$ if and only if $x\; leq\; y$, since in the latter case indeed $x$ is a lower bound of $x$ and $y$, and since clearly $x$ is the "greatest" 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 ("A",$land$) is a meet-semilattice, and the partial order $leq$ is defined as in the universal algebra approach, and $z\; =\; x\; land\; y$ for some elements $x$ and $y$ in "A", then $z$ is the greatest lower bound of $x$ and $y$ with respect to $leq$, since $z\; land\; x\; =\; x\; land\; z\; =\; (x\; land\; x)\; land\; y\; =\; z\; ;Rightarrow;\; z\; leq\; x$, similarly $z\; leq\; y$, and if $w$ is another lower bound of $x$ and $y$, then $w\; land\; x\; =\; w\; land\; y\; =\; w$, whence $w\; land\; z\; =\; w\; land\; (x\; land\; y)\; =\; (w\; land\; x)\; land\; y\; =\; w\; land\; y\; =\; w$. 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 fulfil the conditions for partial orders or meets, respectively.

Meets of general subsets

If ("A",$land$) 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 "A" 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, whence either may be taken as a definition of meet. In the case where "each" subset of "A" has a meet, in fact ("A",$leq$) is a complete lattice; for details, see completeness (order theory).

ee also

*Infima in partially ordered sets
*Meet-semilattice
*Lattice (order) (with extensive references)
*Partially ordered set
*Join (mathematics)