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.

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_ta È d_tb,

(p2) d_t(a Ç b)=d_ta Ç d_tb,

(p3) d_wd_ta=d_ta,

(p4) d_te_s= 1 if tÎ s, otherwise d_te_s=0

(p5) d_ta È Ø d_ta=1

(p6) a = È {e_{s(t) Ç} d_ta: 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_ta, 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_P, È , Ç , ® , Ø , 1, 0) is a Boolean algebra.

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_P, È , Ç , ® , Ø , 1, 0) is a Boolean algebra

(d) For every aÎ C_P, d_tØ a=Ø d_ta, 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_ta = (B_P)È {d_ta_j: jÎ J}

(b) a = (P)Ç {a_j: jÎ J} iff for every tÎ T d_ta = (B_P)Ç {d_ta_j: jÎ J}.

Proposition 1.4

(a) d_t(a® c) = Ç {d_wa® d_wc: w £ t}

(b) d_tØ a = Ç {Ø d_wa: w £ t}

(c) d_wa£ d_ta whenever w£ t, for any w, tÎ T

(d) a£ b iff d_ta£ d_tb 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_wa® d_wc: 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_ts=T if tÎ s, otherwise d_ts=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_wb=(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) is isomorphic to B.

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.

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={e₀,..., 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_ie_j=1 if i£ j, otherwise d_ie_j=0.

Proposition 2.1

(a) e_0® a=e_m-1

(b) e_s® a=È {d_taÇ e_t: t< s}È d_sa

(c) e_m-1® a=a

(d) a® e_s=e_{sÈ Ø} d_s+1a, 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) c₀a=Ø d₁a=Ø a

(c2) c_sa=d_saÇ Ø d_s+1a for sÎ T-{m-1}

(c3) c_m-1a=d_m-1a.

We clearly have

c_saÇ c_ta=e₀ for s¹ t.

Any element a of P has the following disjoint representation:

(c4) a=È {c_taÇ 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 C_P corresponding to a Post algebra of order m coincide.

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).

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 C_P corresponding to a Post algebra of order w ⁺ coincide.

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).

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=e₀ and 1=e_-1, and the following

(p7) d₁a = d_-1a È È {d_sa Ç Ø d_s+1a: 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_te = 1) for all positive t

and e < e_t (d_te =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 C_P corresponding to a Post algebra of order w +w ^* coincide.

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_taÇ e_t+1: tÎ E}

pa=È {c_taÇ 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_taÇ 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_iaÇ c_jb: 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_iaÇ c_jb: 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 e₀ and the ring unit is e₁.

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 S₁ Í S₂ 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.

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