Re: Groups, Magmas, Cayley Tables and Venn Diagrams...

In article <45b0208b-1226-4068-88cd-e4d395891c81@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
landspeedrecord <landspeedrecord@xxxxxxxxx> wrote:
"See the picture placing magmas, quasigroups, semigroups, loops,
monoids, and groups in a lattice of inclusion at the Wikipedia page on
"Magma (algebra)". "

--- Ironically it was precisely this diagram that got me yearning for
a venn diagram instead. The diagram didn't clear things up for me, it
just made me more confused! My difficulty is this: If all these
varying "group-like structures" can all be completely
"represented" (in a loose sense) by a multiplication table then there
must be some sort of venn diagram-esque nesting/intersection of ALL
the different types. All my questions have been asked to clarify this
point... (and you have been most helpful in aiding me to sort it all
out!)

A lattice diagram provides inclusions, from which you can construct a
venn diagram; in fact, since venn diagrams tend to get more and more
complicated the more sets that are involved, when possible lattice
diagrams tend to be far easier to digest and process.

The diagram you advise me to refer to is a good example of my personal
cognitive dissonance in regards to this... It bifurcates instead of
showing how Semi-groups and Quasi-groups are related in the overall
picture.

The "bifurcation" means that (i) neither semigroups nor quasigroups
are a subset of the other; and (ii) the least collection (among the
ones considered) that contains both is the collection of magmas.

Likewise, the "bifurcation" above Groups means that neither loops nor
monoids are subsets of one another, but the intersection (that is, all
loops that are also monoids) is precisely the set of all groups.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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