This post is a continuation of Weirdos but algebraic.

Logically, the quoted post could follow the present one rather than precede it.

Question Does there exist an indecomposable weirdo which is neither an Abelian group nor is based on a module over certain left and right rings, with a distinguished invertible element for each of these rings?

These weirdos were described in my previous post. Also, indecomposable means indecomposable into a direct (Cartesian) product of two nontrivial weirdos.

=====================================

Examples of indecomposable weirdos:

• every weirdo such that the number of its elements is a prime number is indecomposable;

• every cyclic Abelian group of finite order $\ p^n,\ $ where $\ p\ $ is a prime, is indecomposable;

• the additive Abelian group $\ \Bbb Z\ $ is indecomposable;

• the set of ring $\ X:=\Bbb Z[\frac 12]\ $ is an indecomposable weirdo with respect to:

$$ \forall_{x\ y\in X}\quad \sigma(x\ y)\ :=\ \frac{x+y}2. $$

==============================================

EDIT

Please. check my "EDIT" from my previous post.

==============================================

==============================================

PS.   Within a day or two, I will remove these two posts of mine about weirdos, and I will leave MO, this time for good.

MO had great potential, I saw it for all these years. It was not fulfilled at all (just the opposite). So be it.

This post is a continuation of Weirdos but algebraic.

Logically, the quoted post could follow the present one rather than precede it.

Question Does there exist an indecomposable weirdo which is neither an Abelian group nor is based on a module over certain left and right rings, with a distinguished invertible element for each of these rings?

These weirdos were described in my previous post. Also, indecomposable means indecomposable into a direct (Cartesian) product of two nontrivial weirdos.

=====================================

Examples of indecomposable weirdos:

• every weirdo such that the number of its elements is a prime number is indecomposable;

• every cyclic Abelian group of finite order $\ p^n,\ $ where $\ p\ $ is a prime, is indecomposable;

• the additive Abelian group $\ \Bbb Z\ $ is indecomposable;

• the set of ring $\ X:=\Bbb Z[\frac 12]\ $ is an indecomposable weirdo with respect to:

$$ \forall_{x\ y\in X}\quad \sigma(x\ y)\ :=\ \frac{x+y}2. $$

==============================================

EDIT

Please. check my "EDIT" from my previous post.

==============================================

==============================================

PS.   Within a day or two, I will leave MO, this time for good.

MO had great potential, I saw it for all these years. It was not fulfilled at all (just the opposite). So be it.

This post is a continuation of Weirdos but algebraic.

Logically, the quoted post could follow the present one rather than precede it.

Question Does there exist an indecomposable weirdo which is neither an Abelian group nor is based on a module over certain left and right rings, with a distinguished invertible element for each of these rings?

These weirdos were described in my previous post. Also, indecomposable means indecomposable into a direct (Cartesian) product of two nontrivial weirdos.

=====================================

Examples of indecomposable weirdos:

• every weirdo such that the number of its elements is a prime number is indecomposable;

• every cyclic Abelian group of finite order $\ p^n,\ $ where $\ p\ $ is a prime, is indecomposable;

• the additive Abelian group $\ \Bbb Z\ $ is indecomposable;

• the set of ring $\ X:=\Bbb Z[\frac 12]\ $ is an indecomposable weirdo with respect to:

$$ \forall_{x\ y\in X}\quad \sigma(x\ y)\ :=\ \frac{x+y}2. $$

This post is a continuation of Weirdos but algebraic.

Logically, the quoted post could follow the present one rather than precede it.

Question Does there exist an indecomposable weirdo which is neither an Abelian group nor is based on a module over certain left and right rings, with a distinguished invertible element for each of these rings?

These weirdos were described in my previous post. Also, indecomposable means indecomposable into a direct (Cartesian) product of two nontrivial weirdos.

=====================================

Examples of indecomposable weirdos:

• every weirdo such that the number of its elements is a prime number is indecomposable;

• every cyclic Abelian group of finite order $\ p^n,\ $ where $\ p\ $ is a prime, is indecomposable;

• the additive Abelian group $\ \Bbb Z\ $ is indecomposable;

• the set of ring $\ X:=\Bbb Z[\frac 12]\ $ is an indecomposable weirdo with respect to:

$$ \forall_{x\ y\in X}\quad \sigma(x\ y)\ :=\ \frac{x+y}2. $$

==============================================

EDIT

Please. check my "EDIT" from my previous post.

==============================================

==============================================

PS.   Within a day or two, I will remove these two posts of mine about weirdos, and I will leave MO, this time for good.

MO had great potential, I saw it for all these years. It was not fulfilled at all (just the opposite). So be it.