abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

, a derivative algebra is an

algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...

of the signature :<''A'', ·, +, ', 0, 1, ^D> where :<''A'', ·, +, ', 0, 1> 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 in e ...

and ^D is a

unary operator In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on ...

, the derivative operator, satisfying the identities: # 0^D = 0 # ''x''^DD ≤ ''x'' + ''x''^D # (''x'' + ''y'')^D = ''x''^D + ''y''^D. x^D is called the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of x. Derivative algebras provide an algebraic abstraction of the derived set operator in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

. They also play the same role for the

modal logic Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...

''wK4'' = ''K'' + ''p''∧?''p'' → ??''p'' that

s play for ordinary

propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...

References

* Esakia, L., ''Intuitionistic logic and modality via topology'', Annals of Pure and Applied Logic, 127 (2004) 155-170 * McKinsey, J.C.C. and Tarski, A., ''The Algebra of Topology'', Annals of Mathematics, 45 (1944) 141-191 Algebras Boolean algebra Topology {{algebra-stub