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Let be a locally compact group and a discrete subgroup (this means that there exists a neighbourhood of the identity element of such that = {}).Then is called a lattice in if in addition there exists a Borel measure on the quotient space / which is finite (i.e. (/) < +) and -invariant (meaning that for any and any open subset / the equality () = is satisfied).
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).
For example, an element of a distributive lattice is meet-prime if and only if it is meet-irreducible, though the latter is in general a weaker property. By duality, the same is true for join-prime and join-irreducible elements. [7] If a lattice is distributive, its covering relation forms a median graph. [8]
An example is the Knaster–Tarski theorem, which states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice. This is easily seen to be a generalization of the above observation about the images of increasing and idempotent functions.
The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any Maltsev variety (of which groups are an example), the lattice of congruences is modular (Kearnes & Kiss 2013).
For an example, the lattice of subgroups of the dihedral group of order 8 is not modular. The smallest non-modular lattice is the "pentagon" lattice N 5 consisting of five elements 0, 1, x, a, b such that 0 < x < b < 1, 0 < a < 1, and a is not comparable to x or to b. For this lattice, x ∨ (a ∧ b) = x ∨ 0 = x < b = 1 ∧ b = (x ∨ a) ∧ b
The join operation in the all terms lattice yields always an instance of the join in the linear terms lattice; for example, the (ground) terms f(a,a) and f(b,b) have the join f(x,x) and f(x,y) in the all terms lattice and in the linear terms lattice, respectively. As the join operations do not in general agree, the linear terms lattice is not ...
Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤ ~ v and v ≤ ~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the same value in every bounded lattice ...