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| * [https://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols < math > < /math >] | | * [https://www.artofproblemsolving.com/Wiki/index.php/LaTeX:Symbols < math > < /math >] |
| * [http://groupprops.subwiki.org/wiki/Special:ActiveUsers Active Users] | | * [http://groupprops.subwiki.org/wiki/Special:ActiveUsers Active Users] |
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| == Inquiry: A missing property ==
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| (I) A [[loop]] is a quasigroup with a neutral element <math>e</math>.
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| (II) A magma is termed [[unipotent magma|unipotent]] iff there is an element <math>e</math> such that <math>x * x = e</math> for all elements <math>x</math>.
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| Does anybody know how such an element <math>e</math> in (II) is termed? I found [[neutral element|''middle neutral element'']] and ''middle identity'', but I don't know if that term is standardized and generally and uniformly used. Note that such an element <math>\in Q</math> is linked to the neutral element of a certain parastrophe of <math>Q</math>, and that it is the only idempotent element of <math>Q</math>. ''Middle neutral'' or ''middle-neutral'' or ''middle identity'' does not seem to be an apt solution, because it simply does not stand in the middle like a [[Middle-associative elements of magma form submagma|middle-associative element]] does. ''Unipotent'' does not seem to be a good solution, because unipotency is a property of a structure, not of an element. In fact, like the additive identity and the multiplicate identity in a ring, it is the subtractive (right) identity of a quasigroup.
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| In some quasigroups, <math>x * x = e</math> is a kind of 2. In German, the multiplicate identity is called ''Einselement'' (i. e. one-element), while the additive identity is the ''Nullelement'' (i. e. zero-element). So <math>x * x = e</math> as a kind of 2 could be the ''Zweielement'' (i. e. two-element).
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| ''Subtractive identity'' seems to me a good solution, better than possible - just theorizing - ''unineutral'', ''idemneutral'', ''idempotral'', ''idemunit'', ''unidentity'' [YOON-, not UN-], ''mnie'' (= '''m'''iddle '''n'''eutral '''i'''dentity '''e'''lement), ''mneutral'', ''ineutral'' or or or. What about ''transneutral'' (or maybe ''exoneutral''), because in <math>x * x = e</math>, the element <math>e</math> is on the other side of the equals sign, and Latin ''trans'' means on the opposite side, beyond?
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| My aim is to express (II) as easy as (I) such as ''A pool is a quasigroup with a subtractive (right) identity <math>e</math>.'' or ''A pool is a quasigroup with an exoneutral element <math>e</math>.'' Thank you very much in advance. --[[User:CJKG|CJKG]] ([[User talk:CJKG|talk]]) 13:19, 8 July 2014 (UTC)
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