monadic algebra
Let $B$ be a Boolean algebra^{}. An existential quantifier operator on $B$ is a function $\exists :B\to B$ such that

1.
$\exists (0)=0$,

2.
$a\le \exists (a)$, where $a\in B$, and

3.
$\exists (a\wedge \exists (b))=\exists (a)\wedge \exists (b)$, where $a,b\in B$.
A monadic algebra is a pair $(B,\exists )$, where $B$ is a Boolean algebra and $\exists $ is an existential quantifier operator.
There is an obvious connection between an existential quantifier operator on a Boolean algebra and an existential quantifier^{} in a first order logic:

1.
A statement $\phi (x)$ is false iff $\exists x\phi (x)$ is false. For example, suppose $x$ is a real number. Let $\phi (x)$ be the statement $x=x+1$. Then $\phi (x)$ is false no matter what $x$ is. Likewise, $\exists \phi (x)$ is always false too.

2.
$\phi (x)$ implies $\exists x\phi (x)$; in other words, if $\exists x\phi (x)$ is false, then so is $\phi (x)$. For example, let $\phi (x)$ be the statement $$, where $x\in \mathbb{R}$. By itself, $\phi (x)$ is neither true nor false. However $\exists x\phi (x)$ is always true.

3.
$\exists x(\phi (x)\wedge \exists x\psi (x))$ iff $\exists x\phi (x)\wedge \exists x\psi (x)$. For example, suppose again $x$ is real. Let $\phi (x)$ be the statement $$ and $\psi (x)$ the statement $x>1$. Then both $\exists x\psi (x)$ and $\exists x\phi (x)$ are true. It is easy to verify the equivalence of the two sentences^{} in this example. Notice that, however, $\exists x(\phi (x)\wedge \psi (x))$ is false.
Remarks

•
One may replace condition 3. above with the following three conditions to get an equivalent^{} definition of an existential quantifier operator:

(a)
$\exists (\exists (a))=\exists (a)$

(b)
$\exists (a\vee b)=\exists (a)\vee \exists (b)$

(c)
$\exists ({(\exists a)}^{\prime})={(\exists a)}^{\prime}$
From this, it is easy to see that $\exists $ is a closure operator^{} on $B$, and that $\exists a$ and ${(\exists a)}^{\prime}$ are both closed under $\exists $.

(a)

•
Like the Lindenbaum algebra of propositional logic^{}, monadic algebra is an attempt at converting first order logic into an algebra so that a logical question may be turned into an algebraic one. However, the existential quantifier operator in a monadic algebra corresponds to existential quantifier applied to formulas^{} with only one variable (hence the name monadic). Formulas with multiple variables, such as ${x}^{2}+{y}^{2}=1$, $x\le y+z$, or ${x}_{i}={x}_{i+1}+{x}_{i+2}$ where $i=0,1,2,\mathrm{\dots}$ require further generalizations^{} to what is known as a polyadic algebra. The notions of monadic and polyadic algebras were introduced by Paul Halmos.
Dual to the notion of an existential quantifier is that of a universal quantifier. Likewise, there is a dual of an existential quantifier operator on a Boolean algebra, a universal quantifier operator. Formally, a universal quantifier operator on a Boolean algebra $B$ is a function $\forall :B\to B$ such that

1.
$\forall (1)=1$,

2.
$\forall (a)\le a$, where $a\in B$, and

3.
$\forall (a\vee \forall (b))=\forall (a)\vee \forall (b)$, where $a,b\in B$.
Every existential quantifier operator $\exists $ on a Boolean algebra $B$ induces a universal quantifier operator $\forall $, given by
$$\forall (a):={(\exists ({a}^{\prime}))}^{\prime}.$$ 
Conversely, every universal quantifier operator induces an existential quantifier by exchanging $\forall $ and $\exists $ in the definition above. This shows that the two operations are dual to one another.
References
 1 P. Halmos, S. Givant, Logic as Algebra, The Mathematical Association of America (1998).
Title  monadic algebra 

Canonical name  MonadicAlgebra 
Date of creation  20130322 17:48:57 
Last modified on  20130322 17:48:57 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  9 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 03G15 
Classification  msc 06E25 
Related topic  QuantifierAlgebra 
Related topic  PolyadicAlgebra 
Defines  existential quantifier operator 
Defines  universal quantifier operator 