Modal algebra

 ( Algebra ) 

In abstract algebra and logic, a modal algebra is a structure $\backslash langle\; A,\backslash land,\backslash lor,-,0,1,\backslash Box\backslash rangle$ such that

• $\backslash langle\; A,\backslash land,\backslash lor,-,0,1\backslash rangle$ is a Boolean algebra,
• $\backslash Box$ is a unary operation on A satisfying $\backslash Box1=1$ and $\backslash Box(x\backslash land\; y)=\backslash Box\; x\backslash land\backslash Box\; y$ for all x, y in A.

Modal algebras provide models of propositional in the same way as Boolean algebras are models of classical logic. In particular, the variety of all modal algebras is the equivalent algebraic semantics of the modal logic K in the sense of abstract algebraic logic, and the lattice of its subvarieties is dually isomorphic to the lattice of normal modal logics.

Stone's representation theorem 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.

A Magari algebra (or diagonalizable algebra) is a modal algebra satisfying $\backslash Box\; (-\backslash Box\; x\; \backslash lor\; x)\; =\; \backslash Box\; x$. Magari algebras correspond to provability logic.

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See also

• Interior algebra
• Heyting algebra

• A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Logic Guides vol. 35, Oxford University Press, 1997.

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