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- Author or Editor: Tarek Sayed Ahmed x

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## Abstract

Fix 2 *< n < ω* and let CA*
_{n}
* denote the class of cyindric algebras of dimension

*n*. Roughly CA

*is the algebraic counterpart of the proof theory of first order logic restricted to the first*

_{n}*n*variables which we denote by

*L*. The variety RCA

_{n}*of representable CA*

_{n}*s reflects algebraically the semantics of*

_{n}*L*. Members of RCA

_{n}*are concrete algebras consisting of genuine*

_{n}*n*-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CA

*has a finite equational axiomatization, RCA*

_{n}*is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CA*

_{n}*substantially richer than that of Boolean algebras, just as much as*

_{n}*L*is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCA

_{ω,ω}*are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever*

_{n}*L*, fails dramatically for

_{ω,ω}*L*even if we allow certain generalized models that are only locallly classical. It is also shown that any class K such that Nr

_{n}*CA*

_{n}*∩ CRCA*

_{ω}

_{n}**S**

*Nr*

_{c}*CA*

_{n}

_{n}_{+3}, where CRCA

*is the class of completely representable CA*

_{n}*s, and*

_{n}**S**

*denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that*

_{c}**S**

*RaCA*

_{d}

_{ω}**S**

*RaCA*

_{c}_{5}is not elementary, where

**S**

*denotes the operation of forming dense subalgebra.*

_{d}## Abstract

For *β* an ordinal, let PEA_{
β
} (SetPEA_{
β
}) denote the class of polyadic equality (set) algebras of dimension *β*. We show that for any infinite ordinal *α*, if *n* <* ω*, the *n*-neat reduct of _{
n
} (regardless of the representability of *f _{a}
*(

*a*) ≠ 0 and

_{∞,ω }. Various notions of representability (such as ‘satisfying the Lyndon conditions’, weak and strong) are lifted from the level of atom structures to that of atomic algebras and are further characterized via special neat embeddings. As a sample, we show that the class of atomic CA

_{ n }s satisfying the Lyndon conditions coincides with the class of atomic algebras in

**ElS**

_{ c }

**Nr**

_{ n }

**CA**

_{ ω }, where

**El**denotes ‘elementary closure’ and

**S**

_{ c }is the operation of forming complete subalgebras.

Let *α* be an infinite ordinal. Let RCA_{
α
} denote the variety of representable cylindric algebras of dimension *α*. Modifying Andréka’s methods of splitting, we show that the variety RQEA_{
α
} of representable quasi-polyadic equality algebras of dimension *α* is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQA_{
α
} of representable quasi-polyadic algebras of dimension *α*. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCA_{
α
} (proved by Andréka) transfer to universal axiomatizations of RQEA_{
α
}. From such results it can be inferred that any algebraizable extension of *L*
_{
ω,ω
} is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEA_{
ω
} by substitutions indexed by transformations coming from a finitely presented subsemigroup of (^{
ω
}
*ω*, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

## Abstract

We prove completeness, interpolation, decidability and an omitting types theorem for certain multi-dimensional modal logics where the states are not abstract entities but have an inner structure. The states will be sequences. Our approach is algebraic addressing varieties generated by complex algebras of Kripke semantics for such logics. The algebras dealt with are common cylindrification free reducts of cylindric and polyadic algebras. For finite dimensions, we show that such varieties are finitely axiomatizable, have the super amalgamation property, and that the subclasses consisting of only completely representable algebras are elementary, and are also finitely axiomatizable in first order logic. Also their modal logics have an *N P* complete satisfiability problem. Analogous results are obtained for infinite dimensions by replacing finite axiomatizability by finite schema axiomatizability.