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More generally, one may define upper bound and least upper bound for any subset of a partially ordered set X, with “real number” replaced by “element of X ”. In this case, we say that X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound in X.
In the instance where for each task, its period is an exact multiple of every other task that has a shorter period, the task set can be thought of as being composed of n harmonic task subsets of size 1 and therefore =, which makes this generalization equivalent to Liu and Layland's least upper bound.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
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 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 ...
If the order on the group is a lattice order, i.e. any two elements have a least upper bound, then it is a lattice-ordered group (shortly l-group, though usually typeset with a script l: ℓ-group). A Riesz group is an unperforated partially ordered group with a property slightly weaker than being a lattice-ordered group.
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
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For any sets X and Y, X join Y, written X ⊕ Y, is defined to be the union of the sets {2n : n ∈ X} and {2m+1 : m ∈ Y}. The Turing degree of X ⊕ Y is the least upper bound of the degrees of X and Y. Thus is a join-semilattice. The least upper bound of degrees a and b is denoted a ∪ b.