3

I have a set of functions $S$ whose domains are of the form $[0,x]\subset \mathbb{R}$. I then have two binary operation $(S,+,-)$. The "$+$" operation gives a semigroup structure: take $f,g\in S$ with $dom(f)=[0,x_f]$ and $dom(g)=[0,x_g]$ Then

$$dom(f+g)=[0,x_f+x_g]$$ and

$$(f+g)(t)=f(t) \quad t\in [0,x_f],$$ $$(f+g)(t)=g(t-x_f) \quad t\in (x_f,x_f+x_g]$$

The "$-$" operation is something like a symmetric subtraction operation resulting in the portion of one function not on the common interval between the two. It is not associative: $$dom(f-g)=[0,\max\{x_f,x_g\}-\min \{x_f,x_g\}]$$

If $x_g\leq x_f$, then $$(f-g)(t)=f(t+x_g)$$ otherwise $$(f-g)=(t)=g(t+x_f)$$

The $+$ operation is not symmetric while the $-$ operation is.

My question is what is the name of such a structure which has a symmetric operation and an associative operation? It is not a semiring because neither operation distributes over the other. Any feedback is welcome.

Thanks

Brian Paden
  • 75

1 Answers1

0

You're unlikely to find a name for such a thing unless you have some special extra relation between the operations.

Two unrelated operations on a set is hardly worth naming, for nearly all sets admit multiple such operations (even dozens, or infinitely many).

It's usually the interplay between operations that makes algebraic structures with multiple operations worth studying (and naming.)

Look carefully at your operations and maybe there is an interesting link after all.

rschwieb
  • 153,510