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Boolean Algebras, Tarski Invariants, and Index Sets

2006
*
Notre Dame Journal of Formal Logic
*

*Tarski*defined a way of assigning to each

*boolean*

*algebra*, B, an

*invariant*inv(B) ∈ In, where In is a

*set*of triples from N, such that two

*boolean*

*algebras*have the same

*invariant*if

*and*only if they are ... For each x ∈ In we define a complexity class Γ x , that could be either Σ n , Π n , Σ n ∧ Π n , or Π ω+1 depending on x,

*and*prove that the

*set*of indices for computable

*boolean*

*algebras*with

*invariant*... We first need to introduce the concepts of dense

*Boolean*

*algebras*

*and*back-

*and*-forth relations. Dense

*Boolean*

*Algebras*We start by defining the

*Tarski*

*invariants*on elements of a

*Boolean*

*Algebra*. ...

##
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Page 585 of Mathematical Reviews Vol. , Issue 93b
[page]

1993
*
Mathematical Reviews
*

Finally, basing the discussion on a fixed but arbitrary Kleene

*algebra*as*index**set*, the author defines notions of dynamic logic*and*dynamic*algebra*. ... Also the deeper theorem of*Tarski*(if the underlying*set*U is infinite*and*@ is a clone such that @ C @ C G; then @ is not finitely generated) is proved. ...##
###
Invariants, Boolean algebras and ACA$_{0}^{+}$

2005
*
Transactions of the American Mathematical Society
*

Our major results are that the existence of elementary equivalence

doi:10.1090/s0002-9947-05-03802-x
fatcat:bqtwcm2arjdivbx5usr2vdps2a
*invariants*for*Boolean**algebras**and*isomorphism*invariants*for dense*Boolean**algebras*are both of the same strength as ACA + 0 . ... The proof begins with an analogous result about these*invariants*on recursive (dense)*Boolean**algebras*coding 0 (ω) . ...*Tarski*showed that these*invariants*characterize all*Boolean**algebras*up to elementary equivalence Theorem 3.5 ( [1949,*Tarski*] ). For all*Boolean**algebras*B*and*A, B ≡ A ⇔ inv(B) = inv(A). ...##
###
POLYADIC BOOLEAN ALGEBRAS

1954
*
Proceedings of the National Academy of Sciences of the United States of America
*

Even if the

doi:10.1073/pnas.40.5.296
pmid:16589476
pmcid:PMC534124
fatcat:vghqqerdf5fwjm27xixknc57cq
*index**set*I is infinite, the interesting functions are those that depend on finitely many co-ordinates only. The*set*A* of all such functions is a*Boolean**algebra*, as before. ... (This concept occurs also in the announcement of the results of*Tarski**and*Thompson.) A monadic*algebra*is a*Boolean**algebra*with a quantifier. ...##
###
Page 3506 of Mathematical Reviews Vol. , Issue 84i
[page]

1984
*
Mathematical Reviews
*

The decidability of the theory of

*Boolean**algebras*was first shown by A.*Tarski*[Bull. Amer. Math. Soc. 55 (1949), 64; errata, ibid. 55 (1949), 1192]. Yu. L. ... 34:03 082 Li, Xiang 84i:03082 The effective immune*sets**and*the program*index**sets*—a generalization*and*application of the recursion theorem. (Chinese. English summary) Chinese J. ...##
###
Computational Real Algebraic Geometry
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

Let (y 1 , : : :, y r )

doi:10.1201/9781420035315.ch33
fatcat:on3snkonknhxpmruozfldyyvg4
*and*(z 1 , : : :, z s ) be two*Tarski*formulas (with free variables y's*and*z's, respectively); then a formula combining*and*by a*Boolean*connective is a*Tarski*formula with free ... I (m) Q (ni+1) (d) Q O(ni) J i (m) Q i>0 (ni+1) (d) SEMI-*ALGEBRAIC**SETS*Every quanti er-free formula composed of polynomial inequalities*and**Boolean*connectives de nes a semi-*algebraic**set*. ...##
###
Page 673 of Mathematical Reviews Vol. 10, Issue 10
[page]

1949
*
Mathematical Reviews
*

New material in the tenth chapter on

*Boolean**algebras*deals with postulate systems,*Boolean*equations,*and**Boolean*o-*algebras*. The eleventh chapter on applications to*set*theory is new. ...*and**Tarski*,*and*a new treatment of measure theory. ...##
###
Page 490 of Mathematical Reviews Vol. 9, Issue 9
[page]

1948
*
Mathematical Reviews
*

The
author proposes a topology for the

*set*of e-homomorphisms of a e-*Boolean**algebra*into a “topological”*Boolean**algebra**and*announces three theorems concerning this topology. I. ... A “topological*Boolean**algebra*” is a closure*algebra*in the sense of McKinsey*and**Tarski*[Ann. of Math. (2) 45, 141-191 (1944); these Rev. 5, 211], except that the author assumes countable additivity; ...##
###
Coalgebraic Completeness-via-Canonicity
[chapter]

2016
*
Lecture Notes in Computer Science
*

L ML builds the free

doi:10.1007/978-3-319-40370-0_11
fatcat:y3p3xpfau5empn4oohzp5fabv4
*boolean**algebra*over the formal expressions ♦a with a ∈ A,*and*then quotients this object by the fully*invariant*equivalence relation (in BA!) ... An important special case of the adjunction Pf U is its restriction Uf P to*boolean**algebras**and**sets*: since prime filters are maximal in*boolean**algebras*-i.e. ultrafilters (hence the ultrafilter functor ...##
###
Boolean valued semantics for infinitary logics
[article]

2022
*
arXiv
*
pre-print

Mansfield showed that it holds for L_∞∞ if one replaces

arXiv:2112.09416v2
fatcat:6anzx3iijverhccgtgwq2jstc4
*Tarski*semantics with*boolean*valued semantics. ... Furthermore we bring to light (or in some cases just revive) several connections between the infinitary logic L_∞ω*and*the forcing method in*set*theory. ... Γ,Γ ′ ,∆*and*∆ ′ denote*sets*of L ∞∞ -formulae of any cardinality, v, w denote*set*-sized sequences of variables, t, u denote*set*-sized sequences of terms,*and*I denotes an*index**set*. ...##
###
Page 938 of Mathematical Reviews Vol. 19, Issue 9
[page]

1958
*
Mathematical Reviews
*

Let J

*and*J be two*index**sets**and*let f*and*g be two single- valued mappings of J*and*J into E, such that the*sets*{f(i), ie T}*and*{g(j), 7 €¢ J} are the same. ... The structure of*invariant*classes is investigated,*and*it is shown that the class of all*invariant*classes has the power of the continuum. ...##
###
Tarski's Influence on Computer Science
[chapter]

2018
*
Studies in Universal Logic
*

*Tarski*had done much work in the 1930s on

*Boolean*

*algebras*, of which

*algebras*of

*sets*

*and*

*algebras*of propositions (up to equivalence) are specific cases. ...

*sets*,

*algebraic*curve display,

*and*robot motion planning. ...

##
###
Tarski's influence on computer science

2006
*
Logical Methods in Computer Science
*

Here surveyed is the work of

doi:10.2168/lmcs-2(3:6)2006
fatcat:2xrm6o3lyzarrajfedl2ntstqm
*Tarski*on the decision procedure for*algebra**and*geometry, the method of elimination of quantifiers, the semantics of formal languages, modeltheoretic preservation theorems ... ,*and**algebraic*logic; various connections of each with computer science are taken up. ...*sets*,*algebraic*curve display,*and*robot motion planning. ...##
###
Page 7439 of Mathematical Reviews Vol. , Issue 97M
[page]

1997
*
Mathematical Reviews
*

Among the applications, the regular-
ized Lindenbaum-

*Tarski*duality shows that the weak extension of*Boolean*logic (i.e., the semantics of PASCAL-like programming languages) is the logic for semilattice-*indexed*... All this extends from*Set*to any*Boolean*topos. For a non-*Boolean*topos, the proofs do not work but perhaps the results do J. R. ...##
###
Definably simple stable groups with finitary groups of automorphisms

2019
*
Journal of Symbolic Logic (JSL)
*

Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps,

doi:10.1017/jsl.2019.29
fatcat:74e5py4nuvcabeynfwnlu2jwrq
*and*allow us to classify definably simple stable groups in the specific case when they admit such ... Acknowledgements The author is thankful to Alexandre Borovik for proposing the topic, for numerous helpful discussions*and*for precious suggestions on how to improve preliminary versions,*and*to Adrien ... This way we define a definable universe U to be a*set*of*sets*closed under taking singletons (one-element*sets*),*Boolean*operations, direct products,*and*projections. ...
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