Namespaces
Variants
Actions

Difference between revisions of "Magari algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (links)
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
m1100201.png
 +
$#A+1 = 107 n = 1
 +
$#C+1 = 107 : ~/encyclopedia/old_files/data/M110/M.1100020 Magari algebra,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''diagonalizable algebra''
 
''diagonalizable algebra''
  
A [[Boolean algebra|Boolean algebra]] enriched with a [[unary operation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100201.png" />. In the so-expanded [[signature]], the Magari algebra is defined by the axioms of Boolean algebra and the following three specific axioms:
+
A [[Boolean algebra|Boolean algebra]] enriched with a [[unary operation]] $  riangle $.  
 +
In the so-expanded [[signature]], the Magari algebra is defined by the axioms of Boolean algebra and the following three specific axioms:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100202.png" />;
+
1) $  riangle 1 = 1 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100203.png" />;
+
2) $  riangle ( x \wedge y ) = riangle x \wedge riangle y $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100204.png" />.
+
3) $  riangle ( C riangle x \lor x ) = riangle x $.
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100205.png" /> denotes complementation and the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100206.png" /> is the greatest element of the Magari algebra with respect to the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100207.png" />. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100208.png" /> is often employed instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m1100209.png" />.
+
Here, $  C $
 +
denotes complementation and the unit $  1 $
 +
is the greatest element of the Magari algebra with respect to the relation $  \leq  $.  
 +
The notation $  \pmb\tau $
 +
is often employed instead of $  riangle $.
  
One sometimes regards the Magari algebra with the dual operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002011.png" />), defined by the axioms:
+
One sometimes regards the Magari algebra with the dual operation $  \pmb\sigma $(
 +
$  \pmb\sigma = C riangle C $),  
 +
defined by the axioms:
  
1') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002012.png" />;
+
1') $  \pmb\sigma 0 = 0 $;
  
2') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002013.png" />;
+
2') $  \pmb\sigma ( x \lor y ) = \pmb\sigma x \lor \pmb\sigma y $;
  
3') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002014.png" />.
+
3') $  \pmb\sigma ( x \wedge C \pmb\sigma x ) = \pmb\sigma x $.
  
Here, the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002015.png" /> is the least element of the Magari algebra. In order to distinguish the Boolean part of a Magari algebra, one writes the Magari algebra as the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002018.png" /> is a Boolean algebra.
+
Here, the zero 0 $
 +
is the least element of the Magari algebra. In order to distinguish the Boolean part of a Magari algebra, one writes the Magari algebra as the pair $  \langle  {\mathbf A, riangle } \rangle $
 +
or $  \langle  {\mathbf A, \pmb\sigma } \rangle $,  
 +
where $  \mathbf A $
 +
is a Boolean algebra.
  
Magari algebras arose [[#References|[a8]]] as an attempt to treat diagonal phenomena (cf. the diagonalization lemma in [[#References|[a15]]], [[#References|[a4]]]) in the formal Peano arithmetic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002020.png" />, or other  "strong enough"  arithmetical theories (cf. [[Arithmetic, formal|Arithmetic, formal]]) in an algebraic manner. Indeed, the Lindenbaum sentence algebra [[#References|[a12]]] of Peano arithmetic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002022.png" />, equipped with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002023.png" /> defined by
+
Magari algebras arose [[#References|[a8]]] as an attempt to treat diagonal phenomena (cf. the diagonalization lemma in [[#References|[a15]]], [[#References|[a4]]]) in the formal Peano arithmetic, $  { \mathop{\rm PA} } $,  
 +
or other  "strong enough"  arithmetical theories (cf. [[Arithmetic, formal|Arithmetic, formal]]) in an algebraic manner. Indeed, the Lindenbaum sentence algebra [[#References|[a12]]] of Peano arithmetic, $  {\mathcal L} { \mathop{\rm PA} } $,  
 +
equipped with $  riangle $
 +
defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002024.png" /></td> </tr></table>
+
$$
 +
riangle \left \| \varphi \right \| = \left \| { { \mathop{\rm Pr} } ( \lceil  \varphi \rceil ) } \right \| ,
 +
$$
  
is an example of a Magari algebra. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002025.png" /> is the Hilbert–Bernays standard provability predicate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002026.png" /> is the Gödel number of the sentence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002028.png" /> means the equivalence class of the arithmetical sentences formally equivalent in Peano arithmetic to the sentence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002029.png" />. Then, the axioms 1)–2) of Magari algebra correspond to the Löb derivability conditions and 3) corresponds to the formalized Löb theorem (cf. [[#References|[a15]]]). The diagonalization lemma is simulated with the following fixed-point theorem: For every Magari algebra and polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002030.png" /> in its signature, with all the occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002031.png" /> inside the scopes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002032.png" />, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002033.png" /> can be solved in this algebra. Moreover, the solution is unique. It is called the fixed point of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002034.png" /> in the given Magari algebra. Thus, considering the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002036.png" />, one can conclude that there is an arithmetical sentence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002037.png" /> such that
+
is an example of a Magari algebra. Here $  { \mathop{\rm Pr} } ( x ) $
 +
is the Hilbert–Bernays standard provability predicate, $  \lceil  \varphi \rceil $
 +
is the Gödel number of the sentence $  \varphi $
 +
and $  \| \varphi \| $
 +
means the equivalence class of the arithmetical sentences formally equivalent in Peano arithmetic to the sentence $  \varphi $.  
 +
Then, the axioms 1)–2) of Magari algebra correspond to the Löb derivability conditions and 3) corresponds to the formalized Löb theorem (cf. [[#References|[a15]]]). The diagonalization lemma is simulated with the following fixed-point theorem: For every Magari algebra and polynomial $  f ( x ) $
 +
in its signature, with all the occurrences of $  x $
 +
inside the scopes of $  riangle $,  
 +
the equation $  x = f ( x ) $
 +
can be solved in this algebra. Moreover, the solution is unique. It is called the fixed point of the polynomial $  f ( x ) $
 +
in the given Magari algebra. Thus, considering the polynomial $  C riangle x $
 +
on $  {\mathcal L} { \mathop{\rm PA} } $,  
 +
one can conclude that there is an arithmetical sentence $  \varphi $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002038.png" /></td> </tr></table>
+
$$
 +
\left \| \varphi \right \| = C riangle \left \| \varphi \right \| = \left \| {\neg { \mathop{\rm Pr} } ( \lceil  \varphi \rceil ) } \right \| ,
 +
$$
  
i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002039.png" />, which leads to Gödel's first incompleteness theorem; cf. [[Gödel incompleteness theorem|Gödel incompleteness theorem]].
+
i.e., $  { \mathop{\rm PA} } \vdash \varphi \leftrightarrow \neg { \mathop{\rm Pr} } ( \lceil  \varphi \rceil ) $,  
 +
which leads to Gödel's first incompleteness theorem; cf. [[Gödel incompleteness theorem|Gödel incompleteness theorem]].
  
Other examples of Magari algebras arise when one considers a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002041.png" /> is a binary relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002042.png" />. Furthermore, denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002043.png" /> the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002044.png" /> endowed with the operation
+
Other examples of Magari algebras arise when one considers a pair $  \langle  {Q,R } \rangle $,  
 +
where $  R $
 +
is a binary relation on $  Q $.  
 +
Furthermore, denote by $  \langle  {Q,R, riangle } \rangle $
 +
the Boolean algebra $  2  ^ {Q} $
 +
endowed with the operation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002045.png" /></td> </tr></table>
+
$$
 +
riangle X = \left \{ {x \in Q } : {\textrm{ for  every  }  y \in CX,  \textrm{ not  }  yRx } \right \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002046.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002047.png" /> is a Magari algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002048.png" /> is an irreflexive transitive relation satisfying the descending chain condition [[#References|[a3]]]. In fact, every finite Magari algebra is isomorphic to an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002049.png" /> with a finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002050.png" /> and irreflexive transitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002051.png" />.
+
where $  X \subseteq Q $.  
 +
Then $  \langle  {Q,R, riangle } \rangle $
 +
is a Magari algebra if $  R $
 +
is an irreflexive transitive relation satisfying the descending chain condition [[#References|[a3]]]. In fact, every finite Magari algebra is isomorphic to an algebra $  \langle  {Q,R, riangle } \rangle $
 +
with a finite $  Q $
 +
and irreflexive transitive $  R $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002052.png" /> be the [[Stone space|Stone space]] [[#References|[a12]]] of the Boolean restriction of a Magari algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002053.png" />, which is a Stone compactum (see [[Boolean algebra|Boolean algebra]]), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002054.png" /> be defined as:
+
Let $  Q $
 +
be the [[Stone space|Stone space]] [[#References|[a12]]] of the Boolean restriction of a Magari algebra $  \mathbf A $,  
 +
which is a Stone compactum (see [[Boolean algebra|Boolean algebra]]), and let $  R \subseteq Q \times Q $
 +
be defined as:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002055.png" /></td> </tr></table>
+
$$
 +
xRy \iff riangle a \in y  \Rightarrow  a \in x  \textrm{ for  every  }  a \in \mathbf A.
 +
$$
  
Then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002056.png" /> is an embedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002057.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002058.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002059.png" /> is transitive and relatively founded, i.e. every non-empty open-and-closed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002060.png" /> contains minimal elements with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002061.png" /> (the representation theorem).
+
Then the mapping $  a \mapsto \{ {x \in Q } : {a \in x } \} $
 +
is an embedding of $  \mathbf A $
 +
into $  \langle  {Q,R, riangle } \rangle $.  
 +
Moreover, $  R $
 +
is transitive and relatively founded, i.e. every non-empty open-and-closed subset of $  Q $
 +
contains minimal elements with respect to $  R $(
 +
the representation theorem).
  
There is also a duality theorem: The [[Category|category]] of Magari algebras with homomorphisms is equivalent to the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002062.png" /> defined as follows. The objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002063.png" /> are the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002065.png" /> is a Stone compactum and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002066.png" /> is a Boolean relation [[#References|[a5]]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002067.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002068.png" /> is transitive and relatively founded; the morphisms, say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002069.png" />, are the continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002070.png" /> satisfying the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002071.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002072.png" /> denotes composition of two relations.)
+
There is also a duality theorem: The [[Category|category]] of Magari algebras with homomorphisms is equivalent to the category $  \mathbf S $
 +
defined as follows. The objects of $  \mathbf S $
 +
are the pairs $  \langle  {X,R } \rangle $,  
 +
where $  X $
 +
is a Stone compactum and $  R $
 +
is a Boolean relation [[#References|[a5]]] on $  X $
 +
such that $  R ^ {- 1 } $
 +
is transitive and relatively founded; the morphisms, say $  f : {\langle  {X _ {1} ,R _ {1} } \rangle } \rightarrow {\langle  {X _ {2} ,R _ {2} } \rangle } $,  
 +
are the continuous mappings $  f : {X _ {1} } \rightarrow {X _ {2} } $
 +
satisfying the equation $  f \circ R _ {1} = R _ {2} \circ f $.  
 +
(Here $  \circ $
 +
denotes composition of two relations.)
  
The operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002073.png" /> defined by the axioms 1')–3') has an interesting topological interpretation: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002074.png" /> be a [[Scattered space|scattered space]], i.e. a topological space in which every non-empty subset contains an isolated point, regarded along with the derived set operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002075.png" /> [[#References|[a6]]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002076.png" /> is a Magari algebra with respect to the axioms 1')–3'). Conversely, if for some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002078.png" /> is a Magari algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002079.png" /> is a topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002080.png" /> so that the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002081.png" /> is scattered with the derived set operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002082.png" />. A duality between Magari algebras in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002083.png" /> and relatively scattered bitopological spaces with the derived set operator has been established in [[#References|[a2]]].
+
The operation $  \pmb\sigma $
 +
defined by the axioms 1')–3') has an interesting topological interpretation: Let $  Q $
 +
be a [[Scattered space|scattered space]], i.e. a topological space in which every non-empty subset contains an isolated point, regarded along with the derived set operator $  \pmb\sigma $[[#References|[a6]]]. Then $  \langle  {2  ^ {Q} , \pmb\sigma } \rangle $
 +
is a Magari algebra with respect to the axioms 1')–3'). Conversely, if for some set $  Q $,
 +
$  \langle  {2  ^ {Q} , \pmb\sigma } \rangle $
 +
is a Magari algebra, then $  \Omega = \{ {CX } : {\pmb\sigma X \subseteq X } \} $
 +
is a topology on $  Q $
 +
so that the space $  \langle  {Q, \Omega } \rangle $
 +
is scattered with the derived set operator $  \pmb\sigma $.  
 +
A duality between Magari algebras in the form $  \langle  {\mathbf A, \pmb\sigma } \rangle $
 +
and relatively scattered bitopological spaces with the derived set operator has been established in [[#References|[a2]]].
  
The entire set of Magari algebras constitutes a variety (see [[Variety of universal algebras|Variety of universal algebras]]), which is generated by its finite members. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002084.png" /> is a generic (or functionally free) algebra of this variety, though it is proved to be non-isomorphic to the free Magari algebra of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002086.png" /> (see [[Free algebraic system|Free algebraic system]]). Moreover, up to isomorphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002087.png" /> is a proper subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002088.png" />. If one departs from the Zermelo–Fraenkel [[Axiomatic set theory|axiomatic set theory]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002089.png" />, one arrives at another generic Magari algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002090.png" />, the Lindenbaum sentence algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002091.png" />. It is not isomorphic to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002092.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002093.png" />, although the Boolean restrictions of the former and latter are isomorphic. Magari algebras that are Lindenbaum algebras with an additional operation related to other theories interpreting arithmetic are of current interest (cf. [[#References|[a14]]], [[#References|[a10]]], [[#References|[a15]]], [[#References|[a13]]]).
+
The entire set of Magari algebras constitutes a variety (see [[Variety of universal algebras|Variety of universal algebras]]), which is generated by its finite members. $  {\mathcal L} { \mathop{\rm PA} } $
 +
is a generic (or functionally free) algebra of this variety, though it is proved to be non-isomorphic to the free Magari algebra of rank $  \aleph _ {0} $,  
 +
$  F _ {\aleph _ {0}  } $(
 +
see [[Free algebraic system|Free algebraic system]]). Moreover, up to isomorphism, $  F _ {\aleph _ {0}  } $
 +
is a proper subalgebra of $  {\mathcal L} { \mathop{\rm PA} } $.  
 +
If one departs from the Zermelo–Fraenkel [[Axiomatic set theory|axiomatic set theory]] $  { \mathop{\rm ZF} } $,  
 +
one arrives at another generic Magari algebra, $  {\mathcal L} { \mathop{\rm ZF} } $,  
 +
the Lindenbaum sentence algebra of $  { \mathop{\rm ZF} } $.  
 +
It is not isomorphic to either $  {\mathcal L} { \mathop{\rm PA} } $
 +
or $  F _ {\aleph _ {0}  } $,  
 +
although the Boolean restrictions of the former and latter are isomorphic. Magari algebras that are Lindenbaum algebras with an additional operation related to other theories interpreting arithmetic are of current interest (cf. [[#References|[a14]]], [[#References|[a10]]], [[#References|[a15]]], [[#References|[a13]]]).
  
Magari algebras are an algebraic interpretation for provability logic, when the modality connective is interpreted by the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002094.png" />; cf. [[Modal logic|Modal logic]]. This interpretation is related to the normal extensions of provability logic, because there exists a natural isomorphism between the lattice of those extensions and the lattice of varieties of Magari algebras. Introducing a new operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002095.png" /> on a Magari algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002096.png" />, defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002097.png" />, and reading <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m11002099.png" />, one arrives at the Grzegorczyk algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020100.png" />. In view of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020101.png" />, one can reduce <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020102.png" /> to the open elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020103.png" />, thereby obtaining the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020105.png" />-pseudo-Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020106.png" />. Finally, one gets the pseudo-Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020107.png" /> [[#References|[a12]]] by deleting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020108.png" /> from the signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020109.png" />. These transfers give rise to the following connections: The lattice of varieties of Magari algebras and the lattice of varieties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020110.png" />-pseudo-Boolean algebras are isomorphic; there are semi-lattice epimorphisms from the lattice of varieties of Magari algebras onto the lattice of varieties of Grzegorczyk algebras and onto the lattice of varieties of pseudo-Boolean algebras. These connections reflect the corresponding connections between the lattices of extensions of provability logic, proof-intuitionistic logic, intuitionistic propositional logic, and the Grzegorczyk system (cf. [[#References|[a11]]], [[#References|[a7]]]).
+
Magari algebras are an algebraic interpretation for provability logic, when the modality connective is interpreted by the operation $  riangle $;  
 +
cf. [[Modal logic|Modal logic]]. This interpretation is related to the normal extensions of provability logic, because there exists a natural isomorphism between the lattice of those extensions and the lattice of varieties of Magari algebras. Introducing a new operation $  \square $
 +
on a Magari algebra $  \mathbf A $,  
 +
defined as $  \square x = x \wedge riangle x $,  
 +
and reading $  \square $
 +
for $  riangle $,  
 +
one arrives at the Grzegorczyk algebra $  \mathbf A  ^  \square  $.  
 +
In view of the equation $  \square riangle x = riangle x $,  
 +
one can reduce $  riangle $
 +
to the open elements of $  \mathbf A  ^  \square  $,  
 +
thereby obtaining the $  riangle $-
 +
pseudo-Boolean algebra $  \mathbf A  ^ { riangle} $.  
 +
Finally, one gets the pseudo-Boolean algebra $  \mathbf A  ^  \circ  $[[#References|[a12]]] by deleting $  riangle $
 +
from the signature of $  \mathbf A  ^ { riangle} $.  
 +
These transfers give rise to the following connections: The lattice of varieties of Magari algebras and the lattice of varieties of $  riangle $-
 +
pseudo-Boolean algebras are isomorphic; there are semi-lattice epimorphisms from the lattice of varieties of Magari algebras onto the lattice of varieties of Grzegorczyk algebras and onto the lattice of varieties of pseudo-Boolean algebras. These connections reflect the corresponding connections between the lattices of extensions of provability logic, proof-intuitionistic logic, intuitionistic propositional logic, and the Grzegorczyk system (cf. [[#References|[a11]]], [[#References|[a7]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bernardi,  "The fixed-point theorem for diagonalizable algebras"  ''Studia Logica'' , '''34''' :  3  (1975)  pp. 239–251</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Bernardi,  P. D'Aquino,  "Topological duality for diagonalizable algebras"  ''Notre Dame J. Formal Logic'' , '''29''' :  3  (1988)  pp. 345–364</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Akademie  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Hajek,  P. Pudlak,  "Metamathematics of first-order arithmetic" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.R. Halmos,  "Algebraic logic" , Chelsea, reprint  (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press &amp; PWN  (1966)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.V. Kuznetsov,  A. Muravitsky,  "On superintuitionistic logics as fragments of proof logic extensions"  ''Studia Logica'' , '''50''' :  1  (1986)  pp. 77–99</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Magari,  "The diagonalizable algebras"  ''Boll. Unione Mat. Ital.'' , '''12'''  (1975)  pp. 117–125  (suppl. fasc 3)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Magari,  "Representation and duality theory for diagonalizable algebras"  ''Studia Logica'' , '''34''' :  4  (1975)  pp. 305–313</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Magari,  "Algebraic logic and diagonal phenomena" , ''Logic Colloquium '82'' , Elsevier  (1984)  pp. 135–144</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Muravitsky,  "Correspondence of proof-intuitionistic logic extensions to proof-logic extensions"  ''Soviet Math. Dokl.'' , '''31''' :  2  (1985)  pp. 345–348  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  H. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , PWN  (1970)  (Edition: Third)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.Yu. Shavrukov,  "A note on the diagonalizable algebras of PA and ZF"  ''Ann. Pure Appl. Logic'' , '''60''' :  1–2  (1993)  pp. 161–173</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C. Smoryński,  "Fixed point algebras"  ''Bull. Amer. Math. Soc. (N.S.)'' , '''6''' :  3  (1982)  pp. 317–356</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  C. Smoryński,  "Self-reference and modal logic" , Springer  (1985)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A. Muravitsky,  "Magari and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020111.png" />-pseudo-Boolean algebras"  ''Siberian Math. J.'' , '''31''' :  4  (1990)  pp. 623–628  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bernardi,  "The fixed-point theorem for diagonalizable algebras"  ''Studia Logica'' , '''34''' :  3  (1975)  pp. 239–251</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Bernardi,  P. D'Aquino,  "Topological duality for diagonalizable algebras"  ''Notre Dame J. Formal Logic'' , '''29''' :  3  (1988)  pp. 345–364</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Grätzer,  "General lattice theory" , Akademie  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Hajek,  P. Pudlak,  "Metamathematics of first-order arithmetic" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.R. Halmos,  "Algebraic logic" , Chelsea, reprint  (1962)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press &amp; PWN  (1966)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A.V. Kuznetsov,  A. Muravitsky,  "On superintuitionistic logics as fragments of proof logic extensions"  ''Studia Logica'' , '''50''' :  1  (1986)  pp. 77–99</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Magari,  "The diagonalizable algebras"  ''Boll. Unione Mat. Ital.'' , '''12'''  (1975)  pp. 117–125  (suppl. fasc 3)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Magari,  "Representation and duality theory for diagonalizable algebras"  ''Studia Logica'' , '''34''' :  4  (1975)  pp. 305–313</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Magari,  "Algebraic logic and diagonal phenomena" , ''Logic Colloquium '82'' , Elsevier  (1984)  pp. 135–144</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A. Muravitsky,  "Correspondence of proof-intuitionistic logic extensions to proof-logic extensions"  ''Soviet Math. Dokl.'' , '''31''' :  2  (1985)  pp. 345–348  (In Russian)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  H. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , PWN  (1970)  (Edition: Third)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  V.Yu. Shavrukov,  "A note on the diagonalizable algebras of PA and ZF"  ''Ann. Pure Appl. Logic'' , '''60''' :  1–2  (1993)  pp. 161–173</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C. Smoryński,  "Fixed point algebras"  ''Bull. Amer. Math. Soc. (N.S.)'' , '''6''' :  3  (1982)  pp. 317–356</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  C. Smoryński,  "Self-reference and modal logic" , Springer  (1985)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  A. Muravitsky,  "Magari and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110020/m110020111.png" />-pseudo-Boolean algebras"  ''Siberian Math. J.'' , '''31''' :  4  (1990)  pp. 623–628  (In Russian)</TD></TR></table>

Revision as of 04:11, 6 June 2020


diagonalizable algebra

A Boolean algebra enriched with a unary operation $ riangle $. In the so-expanded signature, the Magari algebra is defined by the axioms of Boolean algebra and the following three specific axioms:

1) $ riangle 1 = 1 $;

2) $ riangle ( x \wedge y ) = riangle x \wedge riangle y $;

3) $ riangle ( C riangle x \lor x ) = riangle x $.

Here, $ C $ denotes complementation and the unit $ 1 $ is the greatest element of the Magari algebra with respect to the relation $ \leq $. The notation $ \pmb\tau $ is often employed instead of $ riangle $.

One sometimes regards the Magari algebra with the dual operation $ \pmb\sigma $( $ \pmb\sigma = C riangle C $), defined by the axioms:

1') $ \pmb\sigma 0 = 0 $;

2') $ \pmb\sigma ( x \lor y ) = \pmb\sigma x \lor \pmb\sigma y $;

3') $ \pmb\sigma ( x \wedge C \pmb\sigma x ) = \pmb\sigma x $.

Here, the zero $ 0 $ is the least element of the Magari algebra. In order to distinguish the Boolean part of a Magari algebra, one writes the Magari algebra as the pair $ \langle {\mathbf A, riangle } \rangle $ or $ \langle {\mathbf A, \pmb\sigma } \rangle $, where $ \mathbf A $ is a Boolean algebra.

Magari algebras arose [a8] as an attempt to treat diagonal phenomena (cf. the diagonalization lemma in [a15], [a4]) in the formal Peano arithmetic, $ { \mathop{\rm PA} } $, or other "strong enough" arithmetical theories (cf. Arithmetic, formal) in an algebraic manner. Indeed, the Lindenbaum sentence algebra [a12] of Peano arithmetic, $ {\mathcal L} { \mathop{\rm PA} } $, equipped with $ riangle $ defined by

$$ riangle \left \| \varphi \right \| = \left \| { { \mathop{\rm Pr} } ( \lceil \varphi \rceil ) } \right \| , $$

is an example of a Magari algebra. Here $ { \mathop{\rm Pr} } ( x ) $ is the Hilbert–Bernays standard provability predicate, $ \lceil \varphi \rceil $ is the Gödel number of the sentence $ \varphi $ and $ \| \varphi \| $ means the equivalence class of the arithmetical sentences formally equivalent in Peano arithmetic to the sentence $ \varphi $. Then, the axioms 1)–2) of Magari algebra correspond to the Löb derivability conditions and 3) corresponds to the formalized Löb theorem (cf. [a15]). The diagonalization lemma is simulated with the following fixed-point theorem: For every Magari algebra and polynomial $ f ( x ) $ in its signature, with all the occurrences of $ x $ inside the scopes of $ riangle $, the equation $ x = f ( x ) $ can be solved in this algebra. Moreover, the solution is unique. It is called the fixed point of the polynomial $ f ( x ) $ in the given Magari algebra. Thus, considering the polynomial $ C riangle x $ on $ {\mathcal L} { \mathop{\rm PA} } $, one can conclude that there is an arithmetical sentence $ \varphi $ such that

$$ \left \| \varphi \right \| = C riangle \left \| \varphi \right \| = \left \| {\neg { \mathop{\rm Pr} } ( \lceil \varphi \rceil ) } \right \| , $$

i.e., $ { \mathop{\rm PA} } \vdash \varphi \leftrightarrow \neg { \mathop{\rm Pr} } ( \lceil \varphi \rceil ) $, which leads to Gödel's first incompleteness theorem; cf. Gödel incompleteness theorem.

Other examples of Magari algebras arise when one considers a pair $ \langle {Q,R } \rangle $, where $ R $ is a binary relation on $ Q $. Furthermore, denote by $ \langle {Q,R, riangle } \rangle $ the Boolean algebra $ 2 ^ {Q} $ endowed with the operation

$$ riangle X = \left \{ {x \in Q } : {\textrm{ for every } y \in CX, \textrm{ not } yRx } \right \} , $$

where $ X \subseteq Q $. Then $ \langle {Q,R, riangle } \rangle $ is a Magari algebra if $ R $ is an irreflexive transitive relation satisfying the descending chain condition [a3]. In fact, every finite Magari algebra is isomorphic to an algebra $ \langle {Q,R, riangle } \rangle $ with a finite $ Q $ and irreflexive transitive $ R $.

Let $ Q $ be the Stone space [a12] of the Boolean restriction of a Magari algebra $ \mathbf A $, which is a Stone compactum (see Boolean algebra), and let $ R \subseteq Q \times Q $ be defined as:

$$ xRy \iff riangle a \in y \Rightarrow a \in x \textrm{ for every } a \in \mathbf A. $$

Then the mapping $ a \mapsto \{ {x \in Q } : {a \in x } \} $ is an embedding of $ \mathbf A $ into $ \langle {Q,R, riangle } \rangle $. Moreover, $ R $ is transitive and relatively founded, i.e. every non-empty open-and-closed subset of $ Q $ contains minimal elements with respect to $ R $( the representation theorem).

There is also a duality theorem: The category of Magari algebras with homomorphisms is equivalent to the category $ \mathbf S $ defined as follows. The objects of $ \mathbf S $ are the pairs $ \langle {X,R } \rangle $, where $ X $ is a Stone compactum and $ R $ is a Boolean relation [a5] on $ X $ such that $ R ^ {- 1 } $ is transitive and relatively founded; the morphisms, say $ f : {\langle {X _ {1} ,R _ {1} } \rangle } \rightarrow {\langle {X _ {2} ,R _ {2} } \rangle } $, are the continuous mappings $ f : {X _ {1} } \rightarrow {X _ {2} } $ satisfying the equation $ f \circ R _ {1} = R _ {2} \circ f $. (Here $ \circ $ denotes composition of two relations.)

The operation $ \pmb\sigma $ defined by the axioms 1')–3') has an interesting topological interpretation: Let $ Q $ be a scattered space, i.e. a topological space in which every non-empty subset contains an isolated point, regarded along with the derived set operator $ \pmb\sigma $[a6]. Then $ \langle {2 ^ {Q} , \pmb\sigma } \rangle $ is a Magari algebra with respect to the axioms 1')–3'). Conversely, if for some set $ Q $, $ \langle {2 ^ {Q} , \pmb\sigma } \rangle $ is a Magari algebra, then $ \Omega = \{ {CX } : {\pmb\sigma X \subseteq X } \} $ is a topology on $ Q $ so that the space $ \langle {Q, \Omega } \rangle $ is scattered with the derived set operator $ \pmb\sigma $. A duality between Magari algebras in the form $ \langle {\mathbf A, \pmb\sigma } \rangle $ and relatively scattered bitopological spaces with the derived set operator has been established in [a2].

The entire set of Magari algebras constitutes a variety (see Variety of universal algebras), which is generated by its finite members. $ {\mathcal L} { \mathop{\rm PA} } $ is a generic (or functionally free) algebra of this variety, though it is proved to be non-isomorphic to the free Magari algebra of rank $ \aleph _ {0} $, $ F _ {\aleph _ {0} } $( see Free algebraic system). Moreover, up to isomorphism, $ F _ {\aleph _ {0} } $ is a proper subalgebra of $ {\mathcal L} { \mathop{\rm PA} } $. If one departs from the Zermelo–Fraenkel axiomatic set theory $ { \mathop{\rm ZF} } $, one arrives at another generic Magari algebra, $ {\mathcal L} { \mathop{\rm ZF} } $, the Lindenbaum sentence algebra of $ { \mathop{\rm ZF} } $. It is not isomorphic to either $ {\mathcal L} { \mathop{\rm PA} } $ or $ F _ {\aleph _ {0} } $, although the Boolean restrictions of the former and latter are isomorphic. Magari algebras that are Lindenbaum algebras with an additional operation related to other theories interpreting arithmetic are of current interest (cf. [a14], [a10], [a15], [a13]).

Magari algebras are an algebraic interpretation for provability logic, when the modality connective is interpreted by the operation $ riangle $; cf. Modal logic. This interpretation is related to the normal extensions of provability logic, because there exists a natural isomorphism between the lattice of those extensions and the lattice of varieties of Magari algebras. Introducing a new operation $ \square $ on a Magari algebra $ \mathbf A $, defined as $ \square x = x \wedge riangle x $, and reading $ \square $ for $ riangle $, one arrives at the Grzegorczyk algebra $ \mathbf A ^ \square $. In view of the equation $ \square riangle x = riangle x $, one can reduce $ riangle $ to the open elements of $ \mathbf A ^ \square $, thereby obtaining the $ riangle $- pseudo-Boolean algebra $ \mathbf A ^ { riangle} $. Finally, one gets the pseudo-Boolean algebra $ \mathbf A ^ \circ $[a12] by deleting $ riangle $ from the signature of $ \mathbf A ^ { riangle} $. These transfers give rise to the following connections: The lattice of varieties of Magari algebras and the lattice of varieties of $ riangle $- pseudo-Boolean algebras are isomorphic; there are semi-lattice epimorphisms from the lattice of varieties of Magari algebras onto the lattice of varieties of Grzegorczyk algebras and onto the lattice of varieties of pseudo-Boolean algebras. These connections reflect the corresponding connections between the lattices of extensions of provability logic, proof-intuitionistic logic, intuitionistic propositional logic, and the Grzegorczyk system (cf. [a11], [a7]).

References

[a1] C. Bernardi, "The fixed-point theorem for diagonalizable algebras" Studia Logica , 34 : 3 (1975) pp. 239–251
[a2] C. Bernardi, P. D'Aquino, "Topological duality for diagonalizable algebras" Notre Dame J. Formal Logic , 29 : 3 (1988) pp. 345–364
[a3] G. Grätzer, "General lattice theory" , Akademie (1978)
[a4] P. Hajek, P. Pudlak, "Metamathematics of first-order arithmetic" , Springer (1993)
[a5] P.R. Halmos, "Algebraic logic" , Chelsea, reprint (1962)
[a6] K. Kuratowski, "Topology" , 1 , Acad. Press & PWN (1966)
[a7] A.V. Kuznetsov, A. Muravitsky, "On superintuitionistic logics as fragments of proof logic extensions" Studia Logica , 50 : 1 (1986) pp. 77–99
[a8] R. Magari, "The diagonalizable algebras" Boll. Unione Mat. Ital. , 12 (1975) pp. 117–125 (suppl. fasc 3)
[a9] R. Magari, "Representation and duality theory for diagonalizable algebras" Studia Logica , 34 : 4 (1975) pp. 305–313
[a10] R. Magari, "Algebraic logic and diagonal phenomena" , Logic Colloquium '82 , Elsevier (1984) pp. 135–144
[a11] A. Muravitsky, "Correspondence of proof-intuitionistic logic extensions to proof-logic extensions" Soviet Math. Dokl. , 31 : 2 (1985) pp. 345–348 (In Russian)
[a12] H. Rasiowa, R. Sikorski, "The mathematics of metamathematics" , PWN (1970) (Edition: Third)
[a13] V.Yu. Shavrukov, "A note on the diagonalizable algebras of PA and ZF" Ann. Pure Appl. Logic , 60 : 1–2 (1993) pp. 161–173
[a14] C. Smoryński, "Fixed point algebras" Bull. Amer. Math. Soc. (N.S.) , 6 : 3 (1982) pp. 317–356
[a15] C. Smoryński, "Self-reference and modal logic" , Springer (1985)
[a16] A. Muravitsky, "Magari and -pseudo-Boolean algebras" Siberian Math. J. , 31 : 4 (1990) pp. 623–628 (In Russian)
How to Cite This Entry:
Magari algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Magari_algebra&oldid=47747
This article was adapted from an original article by A. Muravitsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article