In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
, a modal algebra is a structure
such that
*
is a
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
,
*
is a unary operation on ''A'' satisfying
and
for all ''x'', ''y'' in ''A''.
Modal algebras provide models of
propositional modal logics in the same way as Boolean algebras are models of
classical logic. In particular, the
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
of all modal algebras is the equivalent algebraic semantics of the modal logic ''K'' in the sense of
abstract algebraic logic, and the
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
of its subvarieties is dually
isomorphic to the lattice of
normal modal logic In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautologies;
* All instances of the Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed under:
* Detachment rule (''modus po ...
s.
Stone's representation theorem
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first hal ...
can be generalized to the
Jónsson–Tarski duality, which ensures that each modal algebra can be
represented as the algebra of admissible sets in a modal
general frame
In logic, general frames (or simply frames) are Kripke frames with an additional structure, which are used to model modal and intermediate logics. The general frame semantics combines the main virtues of Kripke semantics and algebraic semantics: ...
.
A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying
. Magari algebras correspond to
provability logic Provability logic is a modal logic, in which the box (or "necessity") operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic.
Examples
...
.
See also
*
Interior algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and or ...
*
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
References
A. Chagrov and M. Zakharyaschev, ''Modal Logic'', Oxford Logic Guides vol. 35, Oxford University Press, 1997.
{{algebra-stub
Modal logic
Boolean algebra