High nonassociativity in order 8 and an associative index estimate

Drápal, Aleš; Valent, Viliam

doi:10.1002/jcd.21632

Cited by 9 publications

(24 citation statements)

References 4 publications

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“…Finally, notice that if

\sum_{i = 1}^{n} a_{i} = n = \sum_{i = 1}^{n} b_{i}

then we immediately obtain (under more general assumptions allowing even negative integers) the inequality 2.4(ii) in .…”

Section: Dvt‐inequality—the Two Dimensional Casementioning

confidence: 71%

“…Now, suppose that the equality holds. Then u v w = = = 0, hence ∑ d = 0 then we immediately obtain (under more general assumptions allowing even negative integers) the inequality 2.4(ii) in [2].…”

Section: Prefacementioning

confidence: 72%

“…In , Drápal and Valent proved that

a (Q) \geq 2 n - i (Q) + (δ_{1} + δ_{2})

, where

i (Q)

is the number of idempotents in

Q

, that is,

i (Q) = false| {x \in Q false| x x = x} false|

δ_{1} = false| {z \in Q false| z x \neq x for all x \in Q} false|

, and

δ_{2} = false| {z \in Q false| x z \neq x for all x \in Q} false|

(Theorem 2.5). This important result is an easy consequence of the inequality

\sum_{i = 1}^{n} false(a_{i}^{2} + b_{i}^{2} + a_{i} b_{i} false) - \sum_{i = 1}^{k} false(a_{i} + b_{i} false) \geq 3 n - 2 k + (r + s),

where

n \geq k \geq 0

a_{1}, …, a_{n}, b_{1}, …, b_{n}

are non‐negative integers such that

\sum a_{i} = n = \sum b_{i}

a_{i} \geq 1

, and

b_{i} \geq 1

for

1 \leq i \leq k

r

is the number of

…”

Section: Prefacementioning

confidence: 99%

“…This important result is an easy consequence of the inequality

\sum_{i = 1}^{n} false(a_{i}^{2} + b_{i}^{2} + a_{i} b_{i} false) - \sum_{i = 1}^{k} false(a_{i} + b_{i} false) \geq 3 n - 2 k + (r + s),

where

n \geq k \geq 0

a_{1}, …, a_{n}, b_{1}, …, b_{n}

are non‐negative integers such that

\sum a_{i} = n = \sum b_{i}

a_{i} \geq 1

, and

b_{i} \geq 1

for

1 \leq i \leq k

r

is the number of

i

with

a_{i} = 0

and

s

is the number of

i

with

b_{i} = 0

(Proposition 2.4(ii)). The lengthy and complicated proof of this Drápal–Valent type (DVT)‐inequality in is based on highly semantically involved insight.…”

Section: Prefacementioning

confidence: 99%

“…, r is the number of i with a = 0 i and s is the number of i with b = 0 i (Proposition 2.4(ii)). The lengthy and complicated proof of this Drápal-Valent type (DVT)inequality in [2] is based on highly semantically involved insight. In this ultrashort note, we present an elementary arithmetical proof of a more general inequality of this type (unfortunately on expenses of brutalist syntax).…”

Section: Prefacementioning

confidence: 99%

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A note on one inequality of Drápal–Valent type

Kepka

Němec

2019

J of Combinatorial Designs

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“…Finally, notice that if

\sum_{i = 1}^{n} a_{i} = n = \sum_{i = 1}^{n} b_{i}

then we immediately obtain (under more general assumptions allowing even negative integers) the inequality 2.4(ii) in .…”

Section: Dvt‐inequality—the Two Dimensional Casementioning

confidence: 71%

Section: Prefacementioning

confidence: 72%

“…In , Drápal and Valent proved that

a (Q) \geq 2 n - i (Q) + (δ_{1} + δ_{2})

, where

i (Q)

is the number of idempotents in

Q

, that is,

i (Q) = false| {x \in Q false| x x = x} false|

δ_{1} = false| {z \in Q false| z x \neq x for all x \in Q} false|

, and

δ_{2} = false| {z \in Q false| x z \neq x for all x \in Q} false|

(Theorem 2.5). This important result is an easy consequence of the inequality

\sum_{i = 1}^{n} false(a_{i}^{2} + b_{i}^{2} + a_{i} b_{i} false) - \sum_{i = 1}^{k} false(a_{i} + b_{i} false) \geq 3 n - 2 k + (r + s),

where

n \geq k \geq 0

a_{1}, …, a_{n}, b_{1}, …, b_{n}

are non‐negative integers such that

\sum a_{i} = n = \sum b_{i}

a_{i} \geq 1

, and

b_{i} \geq 1

for

1 \leq i \leq k

r

is the number of

…”

Section: Prefacementioning

confidence: 99%

“…This important result is an easy consequence of the inequality

\sum_{i = 1}^{n} false(a_{i}^{2} + b_{i}^{2} + a_{i} b_{i} false) - \sum_{i = 1}^{k} false(a_{i} + b_{i} false) \geq 3 n - 2 k + (r + s),

where

n \geq k \geq 0

a_{1}, …, a_{n}, b_{1}, …, b_{n}

are non‐negative integers such that

\sum a_{i} = n = \sum b_{i}

a_{i} \geq 1

, and

b_{i} \geq 1

for

1 \leq i \leq k

r

is the number of

i

with

a_{i} = 0

and

s

is the number of

i

with

b_{i} = 0

(Proposition 2.4(ii)). The lengthy and complicated proof of this Drápal–Valent type (DVT)‐inequality in is based on highly semantically involved insight.…”

Section: Prefacementioning

confidence: 99%

Section: Prefacementioning

confidence: 99%

See 3 more Smart Citations