Join and meet
Encyclopedia
In mathematics
, join and meet are dual
binary operation
s on the elements of a partially ordered set
. A join on a set is defined as the (necessarily unique) supremum
(least upper bound) with respect to a partial order on the set, provided a supremum exists. A meet on a set is defined as the unique infimum
(greatest lower bound) with respect to a partial order on the set, provided an infimum exists. If the join of two elements with respect to a given partial order exists then it is always the meet of the two elements in the inverse order, and vice versa.
Usually, the join of two elements x and y is denoted
and the meet of x and y is denoted
Join and meet can be abstractly defined as commutative and associative binary operations satisfying an idempotency law. The two definitions yield equivalent results, except that in the partial order approach it may be possible directly to define joins and meets of more general sets of elements.
A partially ordered set where the join of any two elements always exists is a join-semilattice. A partially ordered set where the meet of any two elements always exists is a meet-semilattice. A partially ordered set where both the join and the meet of any two elements always exist is a lattice
. Lattices provide the most common context in which to find join and meet. In the study of complete lattice
s, the join and meet operations are extended to return the least upper bound and greatest lower bound of an arbitrary set of elements.
In the following we dispense discussing joins, because they become meet when considering the reverse partial order, thanks to duality
.
If there is a meet of x and y, then indeed it is unique, since if both z and z′ are greatest lower bounds of x and y, then and , whence indeed z = z′. If the meet does exist, it is denoted .
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,
∧ on a set A is a meet, if it satisfies the three conditions a, b, and c. The pair (A,∧) then is a meet-semilattice. Moreover, we then may define a binary relation
≤ on A, by stating that if and only if x ∧ y = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,
Note that 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).
, such that each pair of elements in A has a meet, then indeed x ∧ y = x if and only if , 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,∧) is a meet-semilattice, and the partial order ≤ is defined as in the universal algebra approach, and z = x ∧ y for some elements x and y in A, then z is the greatest lower bound of x and y with respect to ≤, since
and therefore . Similarly, , and if w is another lower bound of x and y, then w ∧ x = w ∧ y = w, 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 fulfil the conditions for partial orders or meets, respectively.
s. 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,≤) is a complete lattice
; for details, see completeness (order theory)
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, join and meet are dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s on the elements of a partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
. A join on a set is defined as the (necessarily unique) supremum
Supremum
In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...
(least upper bound) with respect to a partial order on the set, provided a supremum exists. A meet on a set is defined as the unique infimum
Infimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
(greatest lower bound) with respect to a partial order on the set, provided an infimum exists. If the join of two elements with respect to a given partial order exists then it is always the meet of the two elements in the inverse order, and vice versa.
Usually, the join of two elements x and y is denoted
- x ∨ y,
and the meet of x and y is denoted
- x ∧ y.
Join and meet can be abstractly defined as commutative and associative binary operations satisfying an idempotency law. The two definitions yield equivalent results, except that in the partial order approach it may be possible directly to define joins and meets of more general sets of elements.
A partially ordered set where the join of any two elements always exists is a join-semilattice. A partially ordered set where the meet of any two elements always exists is a meet-semilattice. A partially ordered set where both the join and the meet of any two elements always exist is a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
. Lattices provide the most common context in which to find join and meet. In the study of complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
s, the join and meet operations are extended to return the least upper bound and greatest lower bound of an arbitrary set of elements.
In the following we dispense discussing joins, because they become meet when considering the reverse partial order, thanks to duality
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
.
Partial order approach
Let A be a set with a partial order ≤, 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:- z ≤ x and z ≤ y (i.e., z is a lower bound of x and y).
- For any w in A, such that and , we have (i.e., z is greater than or equal to 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′ are greatest lower bounds of x and y, then and , whence indeed z = z′. If the meet does exist, it is denoted .
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 ∧ y = y ∧ x (commutativityCommutativityIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
), - b. x ∧ (y ∧ z) = (x ∧ y) ∧ z (associativityAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
), and - c. x ∧ x = x (idempotency).
Universal algebra approach
By definition, a binary operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
∧ on a set A is a meet, if it satisfies the three conditions a, b, and c. The pair (A,∧) then is a meet-semilattice. Moreover, we then may define a binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
≤ on A, by stating that if and only if x ∧ y = x. In fact, this relation is a partial order on A. Indeed, for any elements x, y, and z in A,
- x ≤ x, since x ∧ x = x by c;
- if x ≤ y and y ≤ x, then x = x ∧ y = y ∧ x = y by a; and
- if x ≤ y and y ≤ z, then x ≤ z, since then x ∧ z = (x ∧ y) ∧ z = x ∧ (y ∧ z) = x ∧ y = x by b.
Note that 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 (A,≤) is a partially ordered setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
, such that each pair of elements in A has a meet, then indeed x ∧ y = x if and only if , 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,∧) is a meet-semilattice, and the partial order ≤ is defined as in the universal algebra approach, and z = x ∧ y for some elements x and y in A, then z is the greatest lower bound of x and y with respect to ≤, since
- z ∧ x = x ∧ z = x ∧ (x ∧ y) = (x ∧ x) ∧ y = x ∧ y = z
and therefore . Similarly, , and if w is another lower bound of x and y, then w ∧ x = w ∧ y = w, whence
- w ∧ z = w ∧ (x ∧ y) = (w ∧ x) ∧ y = w ∧ 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,∧) 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 operationIterated binary operation
In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through repeated application. Common examples include the extension of the addition operation to the summation operation, and the extension of the...
s. 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,≤) is a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
; for details, see completeness (order theory)
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
.