subcommutative
A semigroup $(S,\cdot )$ is said to be left subcommutative if for any two of its elements $a$ and $b$, there exists its element $c$ such that
$ab=ca.$ | (1) |
A semigroup $(S,\cdot )$ is said to be right subcommutative if for any two of its elements $a$ and $b$, there exists its element $d$ such that
$ab=bd.$ | (2) |
If $S$ is both left subcommutative and right subcommutative, it is subcommutative.
The commutativity is a special case of all the three kinds of subcommutativity.
Example 1. The following operation table defines a right subcommutative semigroup $\{0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3}\}$ which is not left subcommutative (e.g. $0\cdot 3=2\ne c\cdot 0$):
$$\begin{array}{ccccc}\hfill \cdot \hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill \mathrm{\hspace{0.33em}0}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill \mathrm{\hspace{0.33em}1}\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill \mathrm{\hspace{0.33em}2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill \mathrm{\hspace{0.33em}3}\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}$$ |
Example 2. The group of the square matrices^{} over a field is both left and right subcommutative (but not commutative^{}), since the equations (1) and (2) are satisfied by
$$c=ab{a}^{-1}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}d={b}^{-1}ab.$$ |
Remark. One uses the above also for a ring $(S,+,\cdot )$ if its multiplicative semigroup $(S,\cdot )$ satisfies the corresponding requirements.
References
- 1 S. Lajos: “On $(m,n)$-ideals in subcommutative semigroups”. – Elemente der Mathematik 24 (1969).
- 2 V. P. Elizarov: “Subcommutative Q-rings”. – Mathematical notes 2 (1967).
Title | subcommutative |
Canonical name | Subcommutative |
Date of creation | 2013-03-22 19:13:45 |
Last modified on | 2013-03-22 19:13:45 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 12 |
Author | pahio (2872) |
Entry type | Definition |
Classification | msc 20M25 |
Classification | msc 20M99 |
Related topic | Commutative |
Related topic | Klein4Ring |
Related topic | Anticommutative |
Defines | left subcommutative |
Defines | right subcommutative |