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Encyclopedia > Monadic Boolean algebra

In abstract algebra, a monadic Boolean algebra is an algebraic structure of the signature

<A, , +, ', 0, 1, ∃>

where

<A, , +, ', 0, 1>

is a Boolean algebra and ∃ is a unary operator, called the existential quantifier, satisfying the identities:

  1. ∃0 = 0
  2. xx
  3. ∃(x + y) = ∃x + ∃y;
  4. xy = ∃(xy)

x is called the existential closure of x. Monadic Boolean algebras play the same role for the monadic logic of quantification that Boolean algebras play for ordinary propositional logic.


The dual of the existential quantifier is the universal quantifier ∀ defined by ∀x = (∃x ' ) '. By the principle of duality, the univeral quantifier satisfies the identities:

  1. ∀1 = 1
  2. xx
  3. ∀(xy) = ∀xy;
  4. x + ∀y = ∀(x + ∀y)

x is called the universal closure of x. The universal quantifier is recoverable from the existential quantifier via the identity ∃x = (∀x ' ) '. Thus the theory of monadic Boolean algebras may be formulated using the universal quantifier instead of the existential. In this formulation one considers algebraic structures of the form <A, , +, ', 0, 1, ∀> where <A, , +, ', 0, 1> is a Boolean algebra and ∀ satisfies the properties of a universal quantifier listed above.


  Results from FactBites:
 
Nov 6 - 10 VU Math Events (797 words)
A classical result about Boolean algebras -- independently proved by Magill, Maxson, and Schein -- states that non-trivial Boolean algebras are isomorphic whenever their endomorphism monoids are isomorphic.
The main point of this talk is to show that the finite part of this classical result is true within monadic Boolean algebras.
By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadic Boolean algebras the endomorphism monoid of each of which has only one element (namely, the identity map).
Boolean Algebras with Operators - STEP - Truman State University (317 words)
Boolean Algebras have been studied for almost 150 years, beginning with G. Boole in the 1850s, and culminating in the famous Stone Representation Theorem in the 1930s, which proves that every Boolean algebra can be thought of as a field of sets.
Boolean algebras with operators (BAOs) are Boolean algebras upon which additional functions with special properties, called operators, have been defined.
Boolean algebras with one additional unary operator have been studied quite a lot in the last 40 years.
  More results at FactBites »

 
 

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