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. $0\cdot 3=2\ne c\cdot 0$):

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.