Neat embeddings as adjoint situations

Sayed-Ahmed, Tarek

doi:10.1007/s11229-013-0344-7

Cited by 3 publications

(4 citation statements)

References 24 publications

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“…If T is strongly rich and if

X_{1}, X_{2}

are subsets of the set of free generators of

{frakturFr}_{ω} V

a \in {Sg}^{{frakturRd}_{qea} A} X_{1}

and

b \in {Sg}^{{frakturRd}_{qea} A} X_{2}

are such that

a \leq b

, then there exists

c \in {Sg}^{frakturA} (X_{1} \cap X_{2})

such that

a \leq c \leq b

. There are countable atomic algebras, when T is rich or strongly rich, that are not completely representable . 3.…”

Section: An Overview and Summary Of Resultsmentioning

confidence: 99%

“…Fix

2 < n < ω

. It is known that

{sans-serifNr}_{n} {CA}_{ω} ⊈ {boldS}_{normald} {sans-serifNr}_{n} {CA}_{ω}

. We do not know whether we can further remove S d proving that any class beween

{sans-serifNr}_{n} {CA}_{ω} \cap {CRCA}_{n}

and

{boldS}_{normalc} {sans-serifNr}_{n} {CA}_{n + 3}

is not elementary.…”

Section: Finite Dimensional Casementioning

confidence: 99%

“…The corresponding logic which is a non‐commutative fragment of Keisler's logic has a ‘Vaught theorem’: Atomic theories have atomic models. Witness for an analogous situation for other fragments of Keisler's logic including itself. Theorem Every atomic CPEA α is completely representable.…”

Section: Cylindric‐polyadic Equality Algebrasmentioning

confidence: 99%

“…One can define a k rounded game

H_{k}

k \leq ω

that is stronger than

G_{k}

(in the sense that for any atomic algebra

frakturC

, ∃ has a winning strategy in

H_{k} (sans-serifAt frakturC) \Rightarrow

∃ has a winning strategyin

G_{k} (sans-serifAt frakturC

)), such that if ∃ has a winning strategy in

H_{ω} (sans-serifAt frakturB)

where

frakturB

is a countable atomic CA m , then

frakturB

is not only completely representable (by ∃'s winning strategy in

G_{ω}

implied by her winning strategy in

H_{ω}

), but using the stronger part of the game involving other moves, ∃ can arrange that

frakturB

satsfies that

At B \in At {sans-serifNr}_{m} {CA}_{ω}

and its Dedekind‐MacNeille completion

Cm At B

is in

{sans-serifNr}_{m} {CA}_{ω}

. (The last two conditions taken together do not imply that

frakturB

itself is in

{sans-serifNr}_{m} {CA}_{ω}

). It can be shown that ∃ has a winning strategy in

H_{k} (sans-serifAt frakturA)

, for all

k < ω

, where

frakturA

is the rainbow algebra based on

double-struckZ

and

double-struckN

used in the previous proof.…”

Section: Finite Dimensional Casementioning

confidence: 99%

See 3 more Smart Citations

On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

Ahmed

2015

Mathematical Logic Qtrly

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We consider countable so‐called rich subsemigroups of (ωω,∘); each such semigroup T gives a variety CPEAT that is axiomatizable by a finite schema of equations taken in a countable subsignature of that of ω‐dimensional cylindric‐polyadic algebras with equality where substitutions are restricted to maps in T. It is shown that for any such T, A∈ CPEA T if and only if frakturA is representable as a concrete set algebra of ω‐ary relations. The operations in the signature are set‐theoretically interpreted like in polyadic equality set algebras, but such operations are relativized to a union of cartesian spaces that are not necessarily disjoint. This is a form of guarding semantics. We show that CPEAT is canonical and atom‐canonical. Imposing an extra condition on T, we prove that atomic algebras in CPEAT are completely representable and that CPEAT has the super amalgamation property. If T is rich and finitely represented, it is shown that CPEAT is term definitionally equivalent to a finitely axiomatizable Sahlqvist variety. Such semigroups exist. This can be regarded as a solution to the central finitizability problem in algebraic logic for first order logic with equality if we do not insist on full fledged commutativity of quantifiers. The finite dimensional case is approached from the view point of guarded and clique guarded (relativized) semantics of fragments of first order logic using finitely many variables. Both positive and negative results are presented.

show abstract

“…If T is strongly rich and if

X_{1}, X_{2}

are subsets of the set of free generators of

{frakturFr}_{ω} V

a \in {Sg}^{{frakturRd}_{qea} A} X_{1}

and

b \in {Sg}^{{frakturRd}_{qea} A} X_{2}

are such that

a \leq b

, then there exists

c \in {Sg}^{frakturA} (X_{1} \cap X_{2})

such that

a \leq c \leq b

. There are countable atomic algebras, when T is rich or strongly rich, that are not completely representable . 3.…”

Section: An Overview and Summary Of Resultsmentioning

confidence: 99%

“…Fix

2 < n < ω

. It is known that

{sans-serifNr}_{n} {CA}_{ω} ⊈ {boldS}_{normald} {sans-serifNr}_{n} {CA}_{ω}

. We do not know whether we can further remove S d proving that any class beween

{sans-serifNr}_{n} {CA}_{ω} \cap {CRCA}_{n}

and

{boldS}_{normalc} {sans-serifNr}_{n} {CA}_{n + 3}

is not elementary.…”

Section: Finite Dimensional Casementioning

confidence: 99%

Section: Cylindric‐polyadic Equality Algebrasmentioning

confidence: 99%

“…One can define a k rounded game

H_{k}

k \leq ω

that is stronger than

G_{k}

(in the sense that for any atomic algebra

frakturC

, ∃ has a winning strategy in

H_{k} (sans-serifAt frakturC) \Rightarrow

∃ has a winning strategyin

G_{k} (sans-serifAt frakturC

)), such that if ∃ has a winning strategy in

H_{ω} (sans-serifAt frakturB)

where

frakturB

is a countable atomic CA m , then

frakturB

is not only completely representable (by ∃'s winning strategy in

G_{ω}

implied by her winning strategy in

H_{ω}

), but using the stronger part of the game involving other moves, ∃ can arrange that

frakturB

satsfies that

At B \in At {sans-serifNr}_{m} {CA}_{ω}

and its Dedekind‐MacNeille completion

Cm At B

is in

{sans-serifNr}_{m} {CA}_{ω}

. (The last two conditions taken together do not imply that

frakturB

itself is in

{sans-serifNr}_{m} {CA}_{ω}

). It can be shown that ∃ has a winning strategy in

H_{k} (sans-serifAt frakturA)

, for all

k < ω

, where

frakturA

is the rainbow algebra based on

double-struckZ

and

double-struckN

used in the previous proof.…”

Section: Finite Dimensional Casementioning

confidence: 99%

See 2 more Smart Citations

On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

Ahmed

2015

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

The class of completely representable polyadic algebras of infinite dimensions is elementary

Ahmed

2014

Algebra Univers.

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Algebraic logic and topoi; a philosophical holistic approach

Ahmed¹

2020

ija

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Neat embeddings as adjoint situations

Cited by 3 publications

References 24 publications

On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

On notions of representability for cylindric‐polyadic algebras, and a solution to the finitizability problem for quantifier logics with equality

The class of completely representable polyadic algebras of infinite dimensions is elementary

Algebraic logic and topoi; a philosophical holistic approach

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