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A Boolean algebra can be interpreted either as a special kind of ring (a Boolean ring) or a special kind of distributive lattice (a Boolean lattice). Each interpretation is responsible for different distributive laws in the Boolean algebra. Similar structures without distributive laws are near-rings and near-fields instead of rings and division ...
Properties (Dl) and (Dr) express biadditivity of φ, which may be regarded as distributivity of φ over addition. Property (A) resembles some associative property of φ. Every ring R is an R-bimodule. So the ring multiplication (r, r′) ↦ r ⋅ r′ in R is an R-balanced product R × R → R.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
An element x is called a dual distributive element if ∀y,z: x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). In a distributive lattice, every element is of course both distributive and dual distributive. In a non-distributive lattice, there may be elements that are distributive, but not dual distributive (and vice versa).
Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division rings, universal enveloping algebras); related structures like rngs; as well as an array of properties that prove to be of interest both within the theory itself and for its applications, such as ...
Rings. Axioms: Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive. Commutative rings. The axioms for rings plus ∀x ∀y xy = yx. Fields. The axioms for commutative rings plus ∀x (¬ x = 0 → ∃y xy = 1) and ¬ 1 = 0.
Because set unions and intersections obey the distributive law, this is a distributive lattice. Birkhoff's theorem states that any finite distributive lattice can be constructed in this way. Theorem. Any finite distributive lattice L is isomorphic to the lattice of lower sets of the partial order of the join-irreducible elements of L.
Draupnir multiplying itself The third gift — an enormous hammer (1902) by Elmer Boyd Smith.The ring Draupnir is visible among other creations by the Sons of Ivaldi.. In Norse mythology, Draupnir (Old Norse: [ˈdrɔupnez̠], "the dripper" [1]) is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night, eight new rings 'drip' from Draupnir, each one of the ...