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By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this ) $$ x*(y\cdot z)=x*y*z\;\; ; \;\; \forall x,y,z\in S? $$

The right and two-sided cases and group-joined-semigroup are defined analogously.

If $\cdot,*$ are two associative binary operations in $S\neq \emptyset$ such that $\cdot=*$ or $x*y=x$ (for all $x,y\in S$) then $(S,\cdot,*)$ is a left semigroup-joined-semigroup, obviously. Now:

(A) Does anyone know some other examples for such algebraic structures?

(B) Any other similar algebraic structures?

(C) Any related topics and/or references?

Example. For every fixed real number $b\neq 0$, the structure $(\mathbb{R},+,+_b)$ satisfies $x+_by+_bz=x+_b(y+_bz)=x+_b(y+z)$, where $x+_by:=x+y-b[\frac{x+y}{b}]$ (see this , this and this).

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    $\begingroup$ (A) If $a\in S,$ and if $x*y=a$ for all $x,y\in S,$ and if $\cdot$ is an associative operation on $S$ such that $\cdot\ne*,$ then $(S,\cdot,*)$ is a "two-sided semigroup-joined-semigroup" different from the examples mentioned in your question. $\endgroup$ – bof Jul 31 '17 at 11:25
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    $\begingroup$ Not a match, but you might benefit from work of Belousov and his student Yuri Movsisyan (one time at Yerevan). They worked on magmas with several binary operations which turned them into quasigroups, and looked at relations involving these operations. I would be mildly surprised if they had never considered a system like what is presented above. Gerhard "Has Been Mildly Surprised Before" Paseman, 2017.07.31. $\endgroup$ – Gerhard Paseman Jul 31 '17 at 17:28
  • $\begingroup$ @ Gerhard Paseman. Thanks, what are their papers/books' titles? $\endgroup$ – M.H.Hooshmand Aug 1 '17 at 9:38
  • $\begingroup$ I do not recall the titles; I did work on hyperidentities and looked at some of their work in this regard. (I have an ArXiv article 1408.2784 which references some of this work.). I suspect you can do a MathSciNet query which will provide a list of article reviews for you to scan. Or you can write to Prof. Movsisyan and ask. Gerhard "It Takes Some Research Effort" Paseman, 2017.08.08. $\endgroup$ – Gerhard Paseman Aug 9 '17 at 5:54
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One example is that $\cdot$ is ordinary multiplication of integers and $x*y=(x\cdot y) \bmod b.$ One could as well let $\cdot$ be ordinary addition. That is essentially your example except that you are not requiring $b$ to be a positive integer.

More generally, Let $\sim$ be an equivalence relation on $S$ and $\cdot$ a compatible associative operation in that $x \cdot y \sim x' \cdot y'$ provided that $x \sim x'$ and $y \sim y'.$ Pick a system of class representatives. Then $x \sim y$ exactly if $[x]=[y]$ and compatibility condition is

$[x\cdot y]=[[x] \cdot [y]]$ for all $x,y.$

Then defining $x*y=[x \cdot y]$ satisfies your requirement since $$x*(y\cdot z)=[x\cdot (y \cdot z)]=[x \cdot (y*z)]=x*(y*z). $$ The middle step since $x \sim x$ and $y \cdot z \sim y*z.$

Moreover $x*y*z=x*(y\cdot z)=(x \cdot y) *z.$ You just have to do $*$ last.

It might be that all examples are like this (but it isn't, see below). Given $*$ and $\cdot$ meeting your condition, define $v \sim w$ to mean that $x*v=x*w$ for all $x.$ This is clearly an equivalence relation and $y\cdot z \sim y*z$ for all $y,z$.

It is not clear that $\cdot$ must be compatible , but it is provided $\cdot$ has a left identity $u$ with $u\cdot v=v$ for all $v.$ Then we can define class representatives by $[x]=u*x $ and we have

$$ [x \cdot y]=u*(x \cdot y)=u*((u \cdot x) \cdot (u \cdot y))=u*(u \cdot x) * (u \cdot y)=u*(u * x) * (u * y) $$ And, $$[[x] \cdot[y]]=u*((u*x)\cdot (u*y))= u*((u*x)* (u*y)).$$

NOTES Having $\sim$ be equality gives the example of $*=\cdot.$ The example in the comments of $x*y=a$ for some constant $a$ is the case that everything in $S$ is equivalent to everything else. I suppose also the case that $x*y=x$ gives the case that everything in $S$ is equivalent to everything else. However here it is not the case that $x*y$ can be described in term of $x\cdot y$

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  • $\begingroup$ Many thanks for your useful example and idea. There is a class of such algebraic structures (namely identical magma-e-magmas) which are completely characterized in several ways, by using decomposer and associative functions. Your idea is near/similar to it (also see sciencedirect.com/science/article/pii/S0019357707800619) $\endgroup$ – M.H.Hooshmand Aug 1 '17 at 8:03

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