MathDB

largest cardinality of a set with a given binary operation

Source: RMM Shortlist 2017 A1

July 4, 2019

algebraBinary operationsetcardinality

Problem Statement

A set

A

is endowed with a binary operation

*

satisfying the following four conditions: (1) If

a, b, c

are elements of

A

, then

a * (b * c) = (a * b) * c

, (2) If

a, b, c

are elements of

a * e = a

such that

a * c = b *c

, then

a = b

, (3) There exists an element

e

of

A

such that

a * e = a

for all

a

in

A

, and (4) If a and b are distinct elements of

A-\{e\}

, then

a^3 * b = b^3 * a^2

, where

x^k = x * x^{k-1}

for all integers

k \ge 2

and all

x

in

A

. Determine the largest cardinality

A

may have.
proposed by Bojan Basic, Serbia

Back to Problems View on AoPS