# 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