semidistributive

Semidistributive is a property that can apply to algebraic structures, particularly in the context of binary operations. An operation denoted by '*' is semidistributive over another operation denoted by '•' if it satisfies certain relationships. Specifically, for any elements a, b, and c in the structure, the semidistributive law states that (a * b) • c = (a • c) * (b • c) or a • (b * c) = (a • b) * (a • c). Some authors may refer to the first as right semidistributive and the second as left semidistributive.

In lattice theory, the semidistributive property is a fundamental concept. A lattice is a partially ordered set where every pair of elements has a unique least upper bound (join) and a unique greatest lower bound (meet). A lattice is called semidistributive if it satisfies the semidistributive law for its join and meet operations. This property is weaker than distributivity. For example, distributive lattices are always semidistributive, but the converse is not true. Non-distributive modular lattices are semidistributive.

The semidistributive property has implications for the structure of the lattice. For instance, in a semidistributive lattice, the absorption laws, a ∨ (a ∧ b) = a and a ∧ (a ∨ b) = a, always hold. However, the distributive laws, a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) and a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), do not necessarily hold. Studying semidistributivity helps to classify and understand various types of lattices beyond the well-known distributive ones.