Derivative algebra (abstract algebra)
Encyclopedia
In abstract algebra
, a derivative algebra is an algebraic structure
of the signature
where
is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities:
xD is called the derivative
of x. Derivative algebras provide an algebraic abstraction of the derived set
operator in topology
. They also play the same role for the modal logic
wK4 = K + p∧□p → □□p that Boolean algebras play for ordinary propositional logic.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, a derivative algebra is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
of the signature
- <A, ·, +, ', 0, 1, D>
where
- <A, ·, +, ', 0, 1>
is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities:
- 0D = 0
- xDD ≤ x + xD
- (x + y)D = xD + yD.
xD is called the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of x. Derivative algebras provide an algebraic abstraction of the derived set
Derived set (mathematics)
In mathematics, more specifically in point-set topology, the derived set of a subset S of a topological space is the set of all limit points of S...
operator in topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. They also play the same role for the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
wK4 = K + p∧□p → □□p that Boolean algebras play for ordinary propositional logic.