POST ALGEBRAS AND POST LOGICS

Ewa Orlowska

Institute of Telecommunications, Warsaw

Emil Post doctoral dissertation Post (1921) contained a description of an n-valued, functionally complete algebra, for a finite n³ 2. The notion of Post algebra was introduced in Rosenbloom (1942). In Rousseau (1969, 1970) an equivalent formulation of the class of Post algebras was given which became a starting point for extensive research. Since then various generalisations of Post algebras inspired by applications in computer science have been developed. This note in a brief survey of major classes of Post algebras.

1. Plain semi-Post algebras

These algebras were introduced and investigated in Cat Ho (1973), Cat Ho and Rasiowa (1987, 1989, 1992). Let (T, £ ) be a poset and let ET be the set of ideals of T together with the empty set Æ . Clearly, TÎ ET. It is known that any sÎ ET is of the form s=È {s(t): s(t) Í s}, where s(t)={wÎ T: w£ t}. The system (ET, Í ) is a complete lattice, where join and meet are set-theoretical union and intersection, respectively.

An abstract algebra

(P) **P**=(P, È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}) where È
, Ç
, ®
are 2-argument operations, Ø
, d_{t} for tÎ
T are unary operations and e_{s} for sÎ
ET are 0-argument operations (constants) is a plain semi-Post algebra (psP-algebra) of type T provided that the following conditions are satisfied:

(p0) (P, È , Ç , ® , Ø )

is a Heyting algebra with the zero element **0**=eÆ
and the unit element **1**=e_{T},

For any a, bÎ P

(p1) d_{t}(a È
b)=d_{t}a È
d_{t}b,

(p2) d_{t}(a Ç
b)=d_{t}a Ç
d_{t}b,

(p3) d_{w}d_{t}a=d_{t}a,

(p4) d_{t}e_{s}= **1 **if tÎ
s, otherwise d_{t}e_{s}=**0**

(p5) d_{t}a È
Ø
d_{t}a=**1**

(p6) a = È
{e_{s(t) Ç
} d_{t}a: tÎ
T} where È
is the least upper bound in P

Let **P**=(P, È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}) be a psP-algebra of type (T, £
).

By B_{P} we denote the set of elements of P of the form d_{t}a, tÎ
T. Then

Proposition 1.1

(b) B_{P} is closed under the operations È
, Ç
, ®
, Ø
of **P**

(c) The algebra **B _{P}**=(B

Let C_{P} be the set of all complemented elements in the distributive lattice (P, È
, Ç
). Then

Proposition 1.2

(b) C_{P} is closed under the operations È
, Ç
, ®
, Ø
of **P**

(c) The algebra **C _{P}**=(C

(d) For every aÎ
C_{P}, d_{tØ
}a=Ø
d_{t}a, tÎ
T.

However, B_{P} and C_{P} do not always equal. Consider a poset (T, £
) such that T={a, b, c} and £
={(b,a)}. Then ET={Æ
, {b}, {c}, {b,c}, {a,b}, T}, B_{P}={Æ
, T}, and C_{P}={Æ
, T, {c}, {a,b}}.

Proposition 1.3 (Epstein lemma)

For any set {a_{j}: jÎ
J} of elements in P it holds

(a) a = (**P**)È
{a_{j}: jÎ
J} iff for every tÎ
T d_{t}a = (**B _{P}**)È
{d

(b) a = (**P**)Ç
{a_{j}: jÎ
J} iff for every tÎ
T d_{t}a = (**B _{P}**)Ç
{d

Proposition 1.4

(a) d_{t}(a®
c) = Ç
{d_{w}a®
d_{w}c: w £
t}

(b) d_{tØ
}a = Ç
{Ø
d_{w}a: w £
t}

(c) d_{w}a£
d_{t}a whenever w£
t, for any w, tÎ
T

(d) a£
b iff d_{t}a£
d_{t}b for all tÎ
T

(e) e_{w£
}e_{t} iff wÍ
t, for any w, tÎ
ET

It follows that every psP-algebra of type (T, £ ) uniquely determines the set

M(P)={ Ç
{d_{w}a®
d_{w}c: w £
t}: tÎ
T} of infinite meets of P.

Observe that for any sets s', s''Î ET there exists the relative pseudo-complement s'® s'' defined by s'® s''=È {sÎ ET: s'Ç sÍ s''}

and the pseudo-complement Ø s' defined by

Ø s'=s'® Æ ==È {sÎ ET: s'Ç s=Æ }

Clearly, s'® s'', Ø s'Î ET.

Proposition 1.5

(a) For any poset (T, £ ), the system (ET, È , Ç , ® , Ø , T, Æ ), where È , Ç are set-theoretical operations of union and intersection, respectively, and ® , Ø are defined as above, is a Heyting algebra with the unit element T and zero element Æ .

(b) Given a psP-algebra **P**=(P, È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}), let EP={e_{s}: sÎ
ET}. Then (EP, £
) is a poset isomorphic to (ET, Í
).

Condition (b) follows from Proposition 1.4(e).

Example 1.1

An important example of a psP-algebra is the following algebra, referred to as a basic psP-algebra:

(ET, È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET})

where (ET, È
, Ç
, ®
, Ø
) is the Heyting algebra defined above and the operations d_{t}, tÎ
T, and e_{s}, sÎ
ET, are defined by:

e_{s}=s, in particular eÆ
=Æ
and e_{T}=T,

d_{t}s=T if tÎ
s, otherwise d_{t}s=Æ
.

Proposition 1.6

The basic psP-algebra is functionally complete, that is any n-argument operation f: ET^{n®
}ET, n=0, 1,...., is definable with the operations of this algebra.

Given a Boolean algebra **B**=(B, È
, Ç
, ®
, -, 1_{B}, 0_{b}) and a poset (T, £
), by a descending T-sequence of elements of B we mean an indexed family (b_{t})_{tÎ
T} of elements of B such that w£
t in T implies b_{t£
}b_{w} in B (for the sake of simplicity we denote the Boolean ordering of B with the same symbol). We say that B and T satisfy condition (erpc) of existence of relative pseudo-complement if

(erpc) For any two descending T-sequences b=(b_{t})_{tÎ
T}, c=(c_{t})_{tÎ
T} of elements of B there exists (**B**)Ç
{b_{w®
}c_{w}: w£
t} for all tÎ
T.

Example 1.2

We present a psP-algebra **P**_{T}(**B**) of type T determined by a Boolean algebra **B**=(B, È
, Ç
, ®
, -, 1_{B}, 0_{B}) such that **B** and T satisfy condition (erpc). The universe P(B) of **P**_{T}(**B**) is the set of all descending T-sequences of elements of B. We define a partial ordering £
on P(B) as follows. Let b=(b_{t})_{tÎ
T} and c=(c_{t})_{tÎ
T} be any elements of P(B). Then

b£
c in P(B) iff b_{t£
}c_{t} in B for all tÎ
T.

The system (P(B), £ ) is a lattice with join and meet defined by

bÈ
c=(b_{tÈ
}c_{t})_{tÎ
T}, bÇ
c=(b_{tÇ
}c_{t})_{tÎ
T}.

Since B and T satisfy (erpc), for any b, c in P(B) there exists the relative pseudo-complement b® c and

b®
c=(x_{t})_{tÎ
T}, where x_{t}=Ç
{b_{w®
}c_{w}: w£
t}.

For every sÎ ET we define

e_{s}=(x_{t})_{tÎ
T}, where x_{t}=1_{B} if tÎ
s, otherwise x_{t}=0_{B}.

Moreover, we put

d_{w}b=(x_{t})_{tÎ
T}, where x_{t}=b_{w} for every tÎ
T,

Ø
b=(x_{t})_{tÎ
T}, where x_{t}=Ç
{-b_{w}: w£
t}.

It is easy to verify that the algebra **P**_{T}(**B**)=(P(B), È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}) defined above is a psP-algebra of type (T, £
).

Proposition 1.7

Let a Boolean algebra **B** and a poset (T, £
) satisfying (erpc) be given. Let **P** be the algebra **P**_{T}(**B**) defined as in Example 1.2. Then the algebra **B _{P} **(see Proposition 1.4)

Example 1.3

A particular instance of the algebra defined in Example 1.2 is a set algebra obtained by taking the field of all subsets of a set as the respective Boolean algebra. Let U be a nonempty set and let B(U) be the field of all subsets of U. We have 1_{B(U)}=U and 0_{B(U)}=Æ
. For any poset (T, £
), B(U) and T satisfy condition (erpc). Let P(B(U)) be the set of all descending T-sequences of sets from B(U). The ordering on P(B(U)) is the inclusion. The algebra **P**_{T}(**B**(U))=(P(B(U)), È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}) defined as in Example 1.2 is a psP-algebra of type (T, £
). The infinite joins in the axiom (p6) are set unions.

Proposition 1.8 (Representation theorem)

Let **P=**(P, È
, Ç
, ®
, Ø
, {d_{t}: tÎ
T}, {e_{s}: sÎ
ET}) be a psP-algebra of type (T, £
). If T is denumerable and either well-founded or the set M(P) is denumerable (in particular if P is denumerable), then for any denumerable set Q of infinite joins and meets in P there exists the field B(U) of all subsets of a nonempty set U and a monomorphism h from **P** into **P**_{T}(**B**(U)) preserving all the operations in Q.

2. Post algebras of order m

The first axiom system for the algebras characterising Post's m-valued logics, for a finite m greater than 2, was presented in Rosenbloom (1942). He called them Post algebras. The axiomatisation was then simplified in Epstein (1960) and Traczyk (1964). Traczyk proved the equational definability of the class of Post algebras. Over the years the theory of Post algebras and several generalisations of these algebras have been developed. Here we define Post algebras of order m as a particular case of psP-algebras.

Let (T_{m}, £
) be a poset such that T_{m}={1,..., m-1}, where m is a natural number greater than 2, and £
is a natural ordering in T_{m}. Then ET_{m}={Æ
, s(1),...,s(m-1)}, where s(t)={wÎ
T_{m}: w£
t}. Clearly, (ET_{m}, Í
) is isomorphic to {0, 1,..., m-1} with the natural ordering. Hence, we can identify these two posets and assume that constants e_{s} are indexed with elements from {0, 1,..., m-1}.

By a Post algebra of order m we mean a psP-algebra of type (T_{m}, £
).

It can be easily shown that this definition is equivalent to the standard definition of Rousseau (1969, 1970).

Example 2.1

A classical example of a Post algebra of order m is an m-element Post algebra such that

P={e_{0},..., e_{m-1}}, and for i, jÎ
{0, 1,..., m-1} the operations in P are defined as follows:

(ex1) e_{iÈ
}e_{j}=e_{max(i,j)},

(ex2) e_{iÇ
}e_{j}=e_{min(i,j)},

(ex3) e_{i®
}e_{j}=**1** if i£
j, otherwise e_{i®
}e_{j}=e_{j},

(ex4) Ø
e_{i}=e_{i®
}**0**,

(ex5) d_{i}e_{j}=**1** if i£
j, otherwise d_{i}e_{j}=**0**.

Proposition 2.1

(a) e_{0®
}a=e_{m-1}

(b) e_{s®
}a=È
{d_{t}aÇ
e_{t}: t<
s}È
d_{s}a

(c) e_{m-1®
}a=a

(d) a®
e_{s}=e_{sÈ
Ø
}d_{s+1}a, for s=0,...,m-2

(e) a®
e_{m-1}=e_{m-1}.

We define disjoint operations c_{s} for sÎ
{0, 1,..., m-1} as follows:

(c1) c_{0}a=Ø
d_{1}a=Ø
a

(c2) c_{s}a=d_{s}aÇ
Ø
d_{s+1}a for sÎ
T-{m-1}

(c3) c_{m-1}a=d_{m-1}a.

We clearly have

c_{s}aÇ
c_{t}a=e_{0} for s¹
t.

Any element a of P has the following disjoint representation:

(c4) a=È
{c_{t}aÇ
e_{t}: tÎ
T}.

Theorems analogous to propositions 1.1,..., 1.8 hold and constructions from examples 1.2 and 1.3 carry over to the case of type (T_{m}, £
). Algebras presented in Example 1.1 can be identified with those defined in Example 2.1. Moreover, the algebras **B _{P} **and

The m-valued Post logic is a propositional logic with binary connectives Ú
, Ù
, ®
, unary connectives Ø
, D_{t} for tÎ
T, and propositional constants E_{s} for sÎ
{0, 1,..., m-1}. The algebraic semantics for the logic is determined in the standard way by the class of Post algebras of order m. A Hilbert-style axiomatisation of m-valued Post logic and its completeness with respect to the algebraic semantics is presented in Rasiowa (1969). The main results on m-valued Post logic include: Model existence theorem (Rasiowa 1970), Craig interpolation theorem (Rasiowa 1970), Herbrand theorem (Perkowska 1972).

Applications of the m-valued Post logic are concerned with the theory of programming. An algorithmic logic based on m-valued Post logic is developed in Perkowska (1972).

Post algebras and logics of any finite type (T, £ ) are considered in Nour (1997). They are also treated in Konikowska, Morgan and Orlowska (1998).

3. Post algebras of order w
^{+}

Let (Tw
, £
) be a poset such that Tw
=w
is the set of natural numbers and £
is the natural ordering of natural numbers. Then ETw
={Æ
, s(1), s(2),..., Tw
}. Clearly, (ETw
, Í
) is isomorphic to {0, 1, 2,...,w
} with the natural ordering. Hence, we can identify these two posets and assume that constants e_{s} are indexed with elements from {0, 1,..., w
}.

A Post algebra of order w
^{+} is a psP-algebra of type (Tw
, £
).

An example of a Post algebra of order w
^{+} can be defined in a way similar to that developed in Example 2.1.

Theorems analogous to propositions 1.1,..., 1.8 hold and constructions from examples 1.2 and 1.3 carry over to the case of type (Tw
, £
). Moreover, the algebras **B _{P} **and

Representation theory for Post algebras of order w
^{+} has been also developed in Maksimova and Vakarelov (1974), Rasiowa (1985).

A Hilbert-style axiomatisation of w
^{+}-valued Post predicate logic and its completeness with respect to the algebraic semantics is presented in Rasiowa (1973). The other results on w
^{+}-valued Post logic include: Kripke style semantics (Maximova and Vakarelov 1974, Vakarelov 1977), Herbrand theorem and a resolution-style proof system (Orlowska 1977, 1978, 1980), relational semantics and a relational proof system (Orlowska 1991).

Applications of the w
^{+}-valued Post logic are concerned with the theory of programming. An algorithmic logic based on w
^{+}-valued Post logic is developed and investigated in Rasiowa (1974a, 1977, 1979).

4. Post algebras of order w
+w
^{*} (in a strict sense)

These algebras are introduced and investigated in Epstein and Rasiowa (1990, 1991). Let T={1, 2,...-2, -1} and E={0, 1, 2,..., -2, -1}. A Post algebra of order w
+w
^{*} is an algebra of the form (P) satisfying axioms (p0),...,(p6), where in (p0) **0**=e_{0} and **1**=e_{-1}, and the following

(p7) d_{1}a = d_{-1}a È
È
{d_{s}a Ç
Ø
d_{s+1}a: 1 £
s <
-1} pivot elimination axiom

(p8) (a®
b)È
(b®
a)=**1**

The axiom (p7) says that an element e such that e_{t} <
e (d_{t}e = **1**) for all positive t

and e <
e_{t} (d_{t}e =**0**) for all negative t does not exist.

Propositions analogous to Propositions1.1, 1.2, 1.3, 1.4 hold for Post algebras of order w
+w
^{*}. Moreover, the algebras **B _{P} **and

Example 4.1

A most natural example of a Post algebra of order w
+w
^{*} is a linear Post algebra of order w
+w
^{*} defined as follows:

P={e_{s}: sÎ
E}, and the operations in P are defined with conditions analogous to (ex1),...,(ex5) from Example 2.1.

Disjoint operations in Post algebras of order w
+w
^{* } can be defined with conditions analogous to (c1), (c2), (c3) from section 2 by replacing m-1 with -1. Then any element a of P has a disjoint representation given by condition (c4).

In Post algebras of order w
+w
^{*} one can define arithmetic-like operations in the following way.

The successor sa (the predecessor pa) of an element a of P is an element given by the following disjoint representation

sa=È
{c_{t}aÇ
e_{t+1}: tÎ
E}

pa=È
{c_{t}aÇ
e_{t-1}: tÎ
E}

provided that either of these exist.

The inverse -a of an element a is given by the disjoint representation

-a=È
{c_{t}aÇ
e_{-1}: tÎ
T} provided that it exists.

Addition and multiplication operations have disjoint representations as follows

a+b=È
{c_{t}(a+b) Ç
e_{t}: tÎ
T} where for each tÎ
T the infinite join c_{t}(a+b)=È
{c_{i}aÇ
c_{j}b: i+j=t} exists

a·
b=È
{c_{t}(a·
b)Ç
e_{t}: tÎ
T} where for each t there is the finite join c_{t}(a·
b)=È
{c_{i}aÇ
c_{j}b: ij=t}.

Proposition 4.1

A Post algebra of order w
+w
^{*} with inverse, addition and multiplication is a commutative ring with unit, where the ring zero is e_{0} and the ring unit is e_{1}.

These rings have the characteristic 0.

For a descending T-sequence X=(X_{t})_{tÎ
T }of subsets from the field B(U) of all subsets of a nonempty set U we define

X^{+}=Ç
{X_{t}: t positive}

X^{-}=È
{X_{t}: t negative}.

It can be shown that the algebra of descending T-sequences X=(X_{t})_{tÎ
T }of sets from B(U) such that X^{+}=X^{-}, with the operations defined as in Example 1.2 is a Post algebra of order w
+w
^{*}.

Representation theorem Post algebras of order w
+w
^{*} has the following form.

Proposition 4.2 (Representation theorem)

For every denumerable Post algebra **P **of order w
+w
^{*} there is a monomorphism h of **P** into a Post set algebra of order w
+w
^{*} whose elements are descending T-sequences X=(X_{t})_{tÎ
T} of sets from the field B(U) of all subsets of a nonempty set U such that X^{+}=X^{-}. Moreover, h preserves a given denumerable set Q of infinite joins and meets of **P**.

Applications of the logic are concerned with approximation reasoning. An approximation reasoning to recognise a subset S of a nonempty universe U is understood as a process of gradual approximating S by

subsets of U S Í
S_{1} Í
S_{2} Í
... which cover S

and subsets ... Í
S_{-2} Í
S_{-1} Í
S which are contained in S.

Then the approximations of set S are defined as follows:

S^{+} = Ç
{S_{t}: t positive}

S^{-} = È
{S_{t}: t negative}.

In Epstein and Rasiowa (1991) a characterisation of sets S such that S^{+} = S^{-} is given.

Post algebras of order u , where u is an arbitrary ordinal number are introduced and investigated in Przymusinska (1980, 1980a, 1980b, 1980c).

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