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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).
A residuated lattice is a lattice. (def) 15. A distributive lattice is modular. [3] 16. A modular complemented lattice is relatively complemented. [4] 17. A boolean algebra is relatively complemented. (1,15,16) 18. A relatively complemented lattice is a lattice. (def) 19. A heyting algebra is distributive. [5] 20. A totally ordered set is a ...
Con(A) is a complete lattice. The compact elements of Con(A) are exactly the finitely generated congruences. Con(A) is an algebraic lattice. Again there is a converse: By a theorem of George Grätzer and E. T. Schmidt, every algebraic lattice is isomorphic to Con(A) for some algebra A.
Atomic lattice may refer to: In mineralogy, atomic lattice refers to the arrangement of atoms into a crystal structure. In order theory, a lattice is called an atomic lattice if the underlying partial order is atomic. In chemistry, atomic lattice refers to the arrangement of atoms in an atomic crystalline solid
The result is a distributive lattice and is used in Birkhoff's representation theorem. However, it may have many more elements than are needed to form a completion of S. [5] Among all possible lattice completions, the Dedekind–MacNeille completion is the smallest complete lattice with S embedded in it. [6]
Hasse diagram of the noncrossing partition lattice on a 4 element set. The leftmost maximal chain is a chief chain. A group is supersolvable if and only if its lattice of subgroups is supersolvable. A chief series of subgroups forms a chief chain in the lattice of subgroups. [3] The partition lattice of a finite set is supersolvable.
Every finite poset is directed-complete and algebraic (though not necessarily bounded-complete). Thus any bounded-complete finite poset is a Scott domain. The natural numbers with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in this direction, see the article on algebraic lattices.
In abstract algebra, a residuated lattice is an algebraic structure that is simultaneously a lattice x ≤ y and a monoid x•y which admits operations x\z and z/y, loosely analogous to division or implication, when x•y is viewed as multiplication or conjunction, respectively. Called respectively right and left residuals, these operations ...
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