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### Boolean Algebras, Tarski Invariants, and Index Sets

Barbara F. Csima, Antonio Montalbán, Richard A. Shore
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.  ...

### 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}^{+}\$

Richard A. Shore
2005 Transactions of the American Mathematical Society
Our major results are that the existence of elementary equivalence 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).  ...

P. R. Halmos
1954 Proceedings of the National Academy of Sciences of the United States of America
Even if the 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]

Bhubaneswar Mishra
2004 Handbook of Discrete and Computational Geometry, Second Edition
Let (y 1 , : : :, y r ) 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]

Fredrik Dahlqvist
2016 Lecture Notes in Computer Science
L ML builds the free 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]

Juan M. Santiago, Matteo Viale
2022 arXiv   pre-print
Mansfield showed that it holds for L_∞∞ if one replaces 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]

Solomon Feferman
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 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

Ulla Karhumäki
2019 Journal of Symbolic Logic (JSL)
Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, 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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