Szerkesztő:MIvanyi/Ferenczi Miklós (matematikus)

   Finite polyadic algebras Finite polyadic algebras are far enough from polyadic algebras. It should be prefixed that finite polyadic algebras are not polyadic algebras whose universum is finite. The terminology finite polyadic algebra is not appropriate enough. It will turn out that finite polyadic algebras and quasi-polyadic algebras, in essential, are the same classes of algebras (Sain-Thompson 1991).

   Polyadic algebras (see Halmos 1962) are far from classical first-order logic. In classical first order logic there are not infinit substitutions. With classical first order logic the so called locally finite infinite dimensional polyadic algebras (Halmos 1962 can be associated. But this class of algebras can not be defined with a finite first order schema.

   Halmos defined a subclass of polyadic algebras, the class of quasi-polyadic algebras (Halmos 1962) which can be defined by a first order axiom schema and which is much closer to classical first order logic than polyadic algebras, in general. But, in the axiomatization occurs a bit unusual transformation sτ, where τ is any finite transformation defined on the dimension (Sági 2012). Sain and Thompson gave a much simpler definition for quasi-polyadic algebras using only an abstract transposition operator pij instead of sτ. They proved (Sain-Thompson 1991) that their definition is term definitional equivalent to Halmos' definition. They called their algebra finite polyadic algebra, see the definition below. So the classes of finite polyadic algebras and quasi-polyadic algebras coincide, in fact.

Definitions

A finitary polyadic equality algebra of dimension α,α>1, or, finitary polyadic algebra, for short (Sain-Thompson 1991) is an algebra
<A,+,.,-,0,1,c_i,s(i/j),p_ij,d_ij>_i,j<α
where + and . binary operations, -, c_i, s(i/j), p_ij, d_ij are unary operations, 0,1 and d_ij are constants, and the axioms below are assumed for every i,j,k<α

(F₀) <A,+,.,-,0,1> is a Boolean algebra, s(i/i)=p_ii=d_ii=IdA and p_ij=p_ji
(F₁) x<c_ix or x=c_ix
(F₂) c_i(x+y)=c_ix+c_iy
(F₃) s(i/j)c_ix=c_ix
(F₄) c_is(i/j)=s(i/j)x, where i is different from j
(F₅) s(i/j)c_kx=c_ks(i/j)x, where k is different from i and j

(F₆) s(i/j) and p_ij are Boolean endomorphisms
(F₇) p_ijp_ijx=x
(F₈) p_ijp_ikx=p_jkp_ijx
(F₉) p_ijs(i/j)x=s(j/i)x
(F₁₀) s(i/j)d_ij=1
(F₁₁) x d_ij<s(i/j)x or equality holds.

   The terminology for the not-Boolean operations are: ith cylindrification (ci), substitution of i by j (s(i/j), abstract transposition of i and j (pij) and the ij diagonal elements (dij), respectively. With ci we can associate the logical meaning ith quantification, with s(i/j) the substitution of the ith variable by the jth one, with pij, the exchange of the ith and jth variables and with dij the formula vi=vj. It is easy to see that the usual Lindenbaum-Tarski algebra associated with classical first order logic and a usual deduction system can be considered as a finite polyadic algebra.
   
   A transposition algebra (Ferenczi 2012) is such a finite polyadic algebra where axiom (F5) is replaced by the following axiom (F5)* axiom
   (F5)*: s(i/k)s(j/m)x= s(j/m)s(i/k)x where the sets {i,k} and {j,m} are disjoint

 Comparing cylindric algebras and finite polyadic algebras (quasi-polyadic algebras), the latter one is, in essentially, a large subclass of cylindric algebras (Sain-Thompson 1991). However, the representation theory of finite polyadic algebras is much more simpler than that of cylindric algebras.
 On the concepts of set algebras with type of the finite polyadic algebras:
 
   
 

   A weak space α U(p) determined a point p in α U and a set U is the set
   {x in α U : x and p are different only in finitely many places}, where U is an arbitrary set.(Leon Henkin, Donald J. Monk and Alfred Tarski (1985)

   A generalized weak transposition set algebra A is a Boolean set algebra whose unit V is an arbitrary union of weak spaces α Uk(p),k in K and A is closed under the set operations ith cylindrification Ci, substitutions S(i/j), transpositions Pij and contains the diagonal elements Dij (i,j<α),furthermore, the disjointness of the weak spaces  α Uk(p) and α Um(p) are assumed if k is different from m (k,m in K).(Ferenczi 2012) The notation of this class is Gwtα.
   A generalized weak relativized-transposition set algebra A is an algebra in Gwtα for which the disjointness condition for the weak spaces is rejected. (Ferenczi 2012) The notation of this class is Gwrtα. 
   It is easy to check that a set algebra in Gwtα is a finite polyadic algebra, while a set algebra in Gwrtα is a transposition algebra (Ferenczi 2013).

   With the classes Gwtα and Gwrtα the classes of cylindric set algebras Gwsα and Gwrsα can be associated (Ferenczi 2013).

Representation

   A finite polyadic algebra is said to be representable if it is isomorphic to a set algebra in Gwtα. A finite polyadic algebra is said to be representable in the relativized sense if it is isomorphic to a set algebra in Gwtα. (Ferenczi 2012)
   
   It is shown that finite polyadic algebras are not representable (Sain-Thompson 1991). The following theorem holds:

   Theorem (Ferenczi 2012) Finite polyadic algebras are relativized representable. In particular, transposition algebras are relativized representable by set algebras in Gwrtα. 
   
   With this theorem the Resek-Thompson representation theorem (including the merry-go-round properties) can be associated (Henkin, Monk and Tarski 1985). A conclusion is that quasi polyadic algebras are also relativised representable.
   

References

• Paul R. Halmos, (1962) Algebraic Logic. Chelsea Publishing
• Ildikó Sain and Richard J. Thompson (1991) Strictly finite schema axiomatization of quasi-polyadic algebras (in Algebraic Logic, eds. H.Andréka, J.D. MOnk and I.Németi, Colloq.Math.Soc.János Bolyai 54, North Holland)
• Miklós Ferenczi (2012) The polyadic generalization of the Boolean axiomatization of fields of sets, Transaction of Amer.Math.Soc., 364, 2, 867-886
• Leon Henkin, J. Donald Monk and Alfred Tarski (1985) Cylindric Algebras Part II., North Holland
• Gábor Sági (2013) Polyadic algebras (in Cylindric-like Algebras and Algebraic Logic, eds. H.Andréka, M.Ferenczi and I. Németi, Bolyai Soc. Math. Studies 22, Springer)
• Miklós Ferenczi (2013) A new representation theory (in Cylindric-like Algebras and Algebraic Logic, eds. H.Andréka, M.Ferenczi and I. Németi, Bolyai Soc. Math. Studies 22, Springer)

Further reading

• Hajnal Andréka, István Németi and Sayed Tarek Ahmed (2013) A non-representable infinit dimensional quasi-polyadic eqaulity algebra with a representable cylindric reduct, Stua Sci. Math. Hun.50. 1, 1-16
• Hajnal Andréka, Leon Henkin, Donald J. Monk and István Németi (1981), Cylindric Set Algebras, Springer