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Science, Math, & Technology

Magmas and Proxies

Posted 1 Month ago by chiarizio

The title will give you no clue what this post is about! Sorry!
This is an abstract Algebra post.
...
A “magma” is any set endowed with a single binary operation under which it is a closed set.
...
Some things that happen in groups, may happen “by proxy” in other magmas.

For instance suppose the set of elements is called S.
And suppose the operation is indicated by juxtaposition. That is, if a and b are both members of S, their product is denoted by ab.

If a is in S and T is a subset of S
then Ta is the set of all ta for all t in T
and aT is the set of all at for all t in T.

A magma S has proxy-left-identities if for every a in S, a is in Sa.
That is, for every a in S there is an x in S (possibly depending on a) such that xa = a.

S has proxy-right-identities if for every a in S, a is in aS.
That is, for every a in S, there is some y in S (possibly depending on a) such that ay = a.

S is proxy-commutative (or commutative-by-proxy) if for every a in S, aS = Sa.
So, for every a in S and every b in S, there exist x in S and y in S, such that xa = ab = by.

S is proxy-associative if for every a in S and every b in S,
a(bS) = (ab)S and
a(Sb) = (aS)b and
S(ab) = (Sa)b.

....

Consider the lattice of all non empty subsets J of S such that SJ is a subset of J and JS is a subset of J.
That is, for every s in S and every j in J, sj is in J and js is in J.

(I need a name for such sets. I’m tempted to call them “ideals”. But that might be a bad idea?)

If S has proxy left identities and proxy right identities and is proxy commutative and proxy associative, then these subsets form a (commutative?) (semigroup?) under the operation JK = { jk | j is in J and k is in K }.
The identity element of this semi group is the set S, which is also the top element of the lattice.
Also, for every such set J and every such set K, JK is a subset of J and JK is a subset of K.

For any element a of S, the set aS = Sa is the smallest such set that contains a as a member. (Still assuming S has proxy identities!)

....

—————. ————- ————- ————— —————

I am going to call an element a of S a “regular” element
if there is an element x of S such that
a(xa) = (ax)a = a and
x(ax) = (xa)x = x and
(ax)(ax) = ax and
(xa)(xa) = xa

Notice ax and xa are both idempotents.
ax acts like a left-identity for a and like a right-identity for x.
xa acts like a left-identity for x and like a right-identity for a.

I am going to call S a “regular” magma if all of its element are regular.

.....

What would someone like to know about these things?

How does S being a regular magma influence the multiplicative lattice of its ideals?

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