In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature
⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,
where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.
The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃):
∃0 = 0
∃x ≥ x
∃(x + y) = ∃x + ∃y
∃x∃y = ∃(x∃y).
∃x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀x := (∃x′)′.
A monadic Boolean algebra has a dual definition and notation that take ∀ as primitive and ∃ as defined, so that ∃x := (∀x′)′. (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra A has signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities:
∀1 = 1
∀x ≤ x
∀(xy) = ∀x∀y
∀x + ∀y = ∀(x + ∀y).
∀x is the universal closure of x.
and 21 Related for: Monadic Boolean algebra information
In abstract algebra, a monadicBooleanalgebra is an algebraic structure A with signature ⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩, where ⟨A, ·, +, ',...
what Booleanalgebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. An interior algebra is an...
algebra MonadicBooleanalgebra De Morgan algebra First-order logic Heyting algebra Lindenbaum–Tarski algebra Skew BooleanalgebraAlgebraic normal form...
In mathematics and mathematical logic, Booleanalgebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the...
(x+y)=\exists x+\exists y} of monadicBooleanalgebra. The axiom (C4) drops out (becomes a tautology). Thus monadicBooleanalgebra can be seen as a restriction...
like the representation theorem for Booleanalgebras and Stone duality fall under the umbrella of classical algebraic logic (Czelakowski 2003). Works in...
Any set of sets closed under the set-theoretic operations forms a Booleanalgebra with the join operator being union, the meet operator being intersection...
logical function), used in logic. Boolean functions are the subject of Booleanalgebra and switching theory. A Boolean function takes the form f : { 0 ...
derive them and operations on them, from first principles Some interactive examples of Church numerals Lambda Calculus Live Tutorial: BooleanAlgebra...
and his students. An elementary version of polyadic algebra is described in monadicBooleanalgebra. In addition to his original contributions to mathematics...
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols[clarification...
Affirming a disjunct Bitwise OR Booleanalgebra (logic) Booleanalgebra topics Boolean domain Boolean function Boolean-valued function Conjunction/disjunction...
Abraham Robinson follows Quine's usage. In philosophy, the adjective monadic is sometimes used to describe a one-place relation such as 'is square-shaped'...
determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers...
done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, compared to Booleanalgebra semantics...
is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic...
Global Information