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

2021
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Bulletin of the Section of Logic
*

We reprove that the class \(\mathsf{CRCA}_n\)

doi:10.18778/0138-0680.2021.17
fatcat:5r6fv4xav5h6xlr7hhnq5yby44
*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*. ...##
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On canonicity and completions of weakly representable relation algebras

2012
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Journal of Symbolic Logic (JSL)
*

AbstractWe show that the variety

doi:10.2178/jsl/1327068701
fatcat:xa5rtl45l5cdlgkyk225xxa43e
*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 ...##
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A polyadic algebra of infinite dimension is completely representable if and only if it is atomic and completely additive
[article]

2013
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arXiv
*
pre-print

We infer, that unlike cylindric

arXiv:1306.1464v1
fatcat:zd575d7ffrf2zfjztuzfhb7y7u
*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*. ...##
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Page 1212 of Mathematical Reviews Vol. , Issue 93c
[page]

1993
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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. ...##
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Page 2443 of Mathematical Reviews Vol. , Issue 2000d
[page]

2000
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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. ...##
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Page 1303 of Mathematical Reviews Vol. , Issue 95c
[page]

1995
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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*. ...##
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A brief history of algebraic logic from neat embeddings to rainbow constructions
[article]

2015
*
arXiv
*
pre-print

The hitherto obtained results generalize seminal results

arXiv:1508.05840v1
fatcat:xsyr345hgzf4natorz4ds6hy6q
*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 ...##
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Squares in Fork Arrow Logic

2003
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Journal of Philosophical Logic
*

In contrast to RAs, FAs are

doi:10.1023/a:1024847106313
fatcat:2fpe5w6f3bchzpy575rogr6yyq
*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. ...##
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Page 270 of Mathematical Reviews Vol. 41, Issue 2
[page]

1971
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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. ...##
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Algebraic analysis of temporal and topological finite variable fragments, using cylindric modal algebras
[article]

2014
*
arXiv
*
pre-print

We show that any class containing the class

arXiv:1402.5998v1
fatcat:3utvq77znfedtbg7ubq5l6xt7q
*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. ...##
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Page 2005 of Mathematical Reviews Vol. , Issue 96d
[page]

1996
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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*...##
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Stone and double Stone algebras: Boolean and Rough Set Representations, 3-valued and 4-valued Logics
[article]

2017
*
arXiv
*
pre-print

ordered tuple

arXiv:1710.02393v1
fatcat:yey4rlgdgvcg5hwywyhnehbe5u
*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: ∨, ∧. ...##
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On neat atom structures for cylindric like algebras
[article]

2013
*
arXiv
*
pre-print

Call an atom structure α weakly k neat

arXiv:1305.5151v1
fatcat:f24asodesbduzo2yirwreexuci
*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*...##
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Page 32 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
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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*. ...

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

2013
*
arXiv
*
pre-print

We show that atomic polyadic

arXiv:1301.5850v1
fatcat:fgit5qnlyfhtxa5itrjy4e4as4
*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. ...
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