What is semilattice in discrete mathematics?
In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset.
Is a partial order?
A partial order defines a notion of comparison. Two elements x and y may stand in any of four mutually exclusive relationships to each other: either x < y, or x = y, or x > y, or x and y are incomparable. A set with a partial order is called a partially ordered set (also called a poset).
What is join in lattice?
There are two binary operations defined for lattices – Join – The join of two elements is their least upper bound. It is denoted by. , not to be confused with disjunction. Meet – The meet of two elements is their greatest lower bound.
What are 2 examples of lattice compounds?
Ionic compounds tend to have high melting and boiling points, because the attraction between ions in the lattice is very strong….Energetics of Ionic Bond Formation.
| Compound | Lattice Energy (kJ/mol) |
|---|---|
| LiF | 1024 |
| SrCl2 | 2130 |
| MgO | 3938 |
Which is the best definition of a semilattice?
Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a meet (or greatest lower bound) for any nonempty finite subset. Every join-semilattice is a meet-semilattice in the inverse order and vice versa.
How is a partial order induced in a semilattice?
A bounded semilattice is an idempotent commutative monoid . A partial order is induced on a meet-semilattice by setting x ≤ y whenever x ∧ y = x. For a join-semilattice, the order is induced by setting x ≤ y whenever x ∨ y = y.
When is a semilattice bounded by an induction argument?
A simple induction argument shows that the existence of all possible pairwise suprema (infima), as per the definition, implies the existence of all non-empty finite suprema (infima). A join-semilattice is bounded if it has a least element, the join of the empty set.
Do you need a monoid for a semilattice?
Traditionally, a semilattice need have only finite inhabited meets/joins; that is, it need not have a top/bottom element. Algebraically, this means that a semilattice need not be a monoid, but is any commutative idempotent semigroup.