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On complete representations and minimal completions in algebraic logic, both positive and negative results

Tarek Sayed Ahmed
2021 Bulletin of the Section of Logic  
We reprove that the class \(\mathsf{CRCA}_n\) of completely representable \(\mathsf{CA}_n\)s, and the class \(\sf CRRA\) of completely representable \(\mathsf{RA}\)s are not elementary, a result of Hirsch  ...  We show that that the class of completely and strongly representable algebras in \(\sf V\) is not elementary either, reproving a result of Bulian and Hodkinson.  ...  When it is only one meet consisting of co-atoms, in an atomic algebra, this representation is a complete one.  ... 
doi:10.18778/0138-0680.2021.17 fatcat:5r6fv4xav5h6xlr7hhnq5yby44

On canonicity and completions of weakly representable relation algebras

Ian Hodkinson, Szabolcs Mikulás
2012 Journal of Symbolic Logic (JSL)  
AbstractWe show that the variety of weakly representable relation algebras is neither canonical nor closed under Monk completions.  ...  It is obtained by dropping the requirement on classical representations of relation algebras that +, − are respected.  ...  Nonetheless, it can still be helpful in a number of ways to know that the canonical extension of a representable algebra is also representable. 1 Several kinds of representable algebra do indeed have  ... 
doi:10.2178/jsl/1327068701 fatcat:xa5rtl45l5cdlgkyk225xxa43e

A polyadic algebra of infinite dimension is completely representable if and only if it is atomic and completely additive [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
We infer, that unlike cylindric algebras, there is a first order axiomatization of the class of completely representable polyadic algebras of infinite dimension, though the one we obtain is infinite; in  ...  Similar results are obtained for non commutative reducts of polyadic equality algebras of infinite dimensions, where we can drop complete additivity.  ...  When it is only one meet consisting of co-atoms, in an atomic algebra, this representation is a complete one.  ... 
arXiv:1306.1464v1 fatcat:zd575d7ffrf2zfjztuzfhb7y7u

Page 1212 of Mathematical Reviews Vol. , Issue 93c [page]

1993 Mathematical Reviews  
The main theorem is the REL representation theorem, 5.4, in which every complete atomic algebra in REL is shown to be isomorphic to a relativization of a representable relation algebra, and hence, since  ...  Using cylindric atom structures, the author proves several results on the (non-) representability of certain classes of cylindric algebras of dimension three.  ... 

Page 2443 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
However, the requirement of atomicity limits the application of these results, e.g., it is not clear how to use the representation theorem to achieve completeness results in logic.  ...  On reducts of Tarski relation algebras. (Russian. Russian summary) Algebra i Logika 37 (1998), no. 1, 3-16, 122; translation in Algebra and Logic 37 (1998), no. 1, 1-8.  ... 

Page 1303 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
A concrete logic that is isomorphic to a set logic E is called a representation of E.  ...  The author studies a set logic in terms of all of its representations together with the automorphism groups of the representations.  ... 

A brief history of algebraic logic from neat embeddings to rainbow constructions [article]

Tarek Sayed Ahmed
2015 arXiv   pre-print
The hitherto obtained results generalize seminal results of Hirsch and Hodkinson on non--atom canonicity, non--first order definabiity and non--finite axiomatizability, proved for classes of representable  ...  We take a long magical tour in algebraic logic, starting from classical results on neat embeddings due to Henkin, Monk and Tarski, all the way to recent results in algebraic logic using so--called rainbow  ...  Such sequences witness the non-finite axiomatizability of the class representable agebras and the elementary closure of the class completely representable ones, namely, the class of algebras satisfying  ... 
arXiv:1508.05840v1 fatcat:xsyr345hgzf4natorz4ds6hy6q

Squares in Fork Arrow Logic

Renata P. de Freitas, Jorge P. Viana, Mario R. F. Benevides, Sheila R. M. Veloso, Paulo A. S. Veloso
2003 Journal of Philosophical Logic  
In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic.  ...  FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs).  ...  Also, the referees' comments have been instrumental in providing the clarity of the presentation.  ... 
doi:10.1023/a:1024847106313 fatcat:2fpe5w6f3bchzpy575rogr6yyq

Page 270 of Mathematical Reviews Vol. 41, Issue 2 [page]

1971 Mathematical Reviews  
The algebraic counterpart of completeness in predicate logic is representability of cylindric algebras. A. Tarski has shown [Nederl. Akad. Wetensch. Proc. Ser.  ...  Donald 1517 Nonfinitizability of classes of representable cylindric algebras. J. Symbolic Logic 34 (1969), 331-343.  ... 

Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras [article]

Tarek Sayed Ahmed
2014 arXiv   pre-print
We show that any class containing the class of completely representable algebras having a weak neat embedding property is not elementary.  ...  The semantics of representable topological algebras is induced by the interior operation relative to a topology defined on their bases.  ...  Lately, it has become fashionable in algebraic logic to study representations of abstract algebras that has a complete representation [34] for an extensive overview.  ... 
arXiv:1402.5998v1 fatcat:3utvq77znfedtbg7ubq5l6xt7q

Page 2005 of Mathematical Reviews Vol. , Issue 96d [page]

1996 Mathematical Reviews  
A complete representation of a Boolean algebra B is an injective complete homomorphism h: B — A(X).  ...  Here we complement this result by a representation theorem: every atomic abstract fork algebra is isomorphic to a proper one.” 96d:03082 03G15 03G05 06E0S Hirsch, Robin (4-LNDIC-C; London) Completely representable  ... 

Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics [article]

Arun Kumar
2017 arXiv   pre-print
ordered tuple of sets. 3-valued semantics of logic for Stone algebra, dual Stone algebras and 4-valued semantics of logic for double Stone algebras are proposed and established soundness and completeness  ...  Further it was proved by Moisil himself the representations of n-valued Łukasiewicz Moisil algebra in terms of B^[n].  ...  The study of logics in this article is completely based on DLL. Let us present the logic. The language consists of -Propositional variables: p, q, r, . . .. -Logical connectives: ∨, ∧.  ... 
arXiv:1710.02393v1 fatcat:yey4rlgdgvcg5hwywyhnehbe5u

On neat atom structures for cylindric like algebras [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
Call an atom structure α weakly k neat representable, the term algebra is in _n∩_n_n+k, but the complex algebra is not representable.  ...  Call an atom structure α k neat if there exists such that =α, and ∈_n_n+k. (4) Let K⊆_n, and Ł be an extension of first order logic.  ...  Journal of Symbolic Logic, 72(2) (2007), p.673-703. [24] Hirsch and Hodkinson complete representations in algebraic logic Journal of Symbolic Logic (1996). [25] Hirsch and Hodkison Completions and complete  ... 
arXiv:1305.5151v1 fatcat:f24asodesbduzo2yirwreexuci

Page 32 of Mathematical Reviews Vol. , Issue 98A [page]

1998 Mathematical Reviews  
Algebraic logic and, in particular, categorical logic replace a completeness theorem by a representa- tion theorem.” A number of known examples are mentioned for different kinds of logics.  ...  98a:03097 03G Algebraic logic 98a:03097 03G15 03-02 Monk, J. Donald (1-CO; Boulder, CO) Lectures on cylindric set algebras.  ... 

Atomic polyadic algebras of infinite dimension are completely representable [article]

Tarek Sayed Ahmed
2013 arXiv   pre-print
We show that atomic polyadic algebras of infinite dimensions are completely representable  ...  of cylindric algebras of Tarski and that of (quasi)polyadic algebras of Halmos, which constitute the most two important algebriazations of first order logic and certain infinitary expansions of it, has  ...  In fact the definability of substitutions in cylindric algebras have initiated a lot of research, we refer to [2] for more references dealing with such a topic.  ... 
arXiv:1301.5850v1 fatcat:fgit5qnlyfhtxa5itrjy4e4as4
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