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

\displaystyle 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

\displaystyle 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 $\{0,\,1,\,2,\,3\}$ which is not left subcommutative (e.g. $0\!\cdot\!3=2\neq c\!\cdot\!0$ ):

\begin{array}[]{c|cccc}\cdot&0&1&2&3\\ \hline\;0&0&0&2&2\\ \;1&0&1&2&3\\ \;2&0&0&2&2\\ \;3&0&1&2&3\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\;=\;aba^{-1}\quad\mbox{and}\quad 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