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
∀x ≤ x
∀(xy) = ∀x∀y;
∀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.
A classical result about Booleanalgebras -- independently proved by Magill, Maxson, and Schein -- states that non-trivial Booleanalgebras 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 monadicBooleanalgebras.
By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadicBooleanalgebras the endomorphism monoid of each of which has only one element (namely, the identity map).
BooleanAlgebras 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 Booleanalgebra can be thought of as a field of sets.
Booleanalgebras with operators (BAOs) are Booleanalgebras upon which additional functions with special properties, called operators, have been defined.
Booleanalgebras with one additional unary operator have been studied quite a lot in the last 40 years.
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