In Basic Algebra 1 by Jacobson, he dosen't use the notation $K$, but he does give what you are talking about as an example. It is a monoid, and not, nessacarilly, a group. A monoid is a structure that satisfies all of the group axioms, except for possibly inverses. So, consider it the notation for a monoid generated by t. Of course, the cyclic group is also a monoid, however.
For a specific example, look at the integers under addition. The monoid generated b 1 would be the natural numbers, with zero. Moreover, this is not a group. However, if you were working in $\mathbb{Z}_n$, under addition. In this case, all inverses can be written as positive 'powers.'
For an example, modulo 6, the inverse of two is 4, which is 2+2 (to the power of two, I know that you know that 2+2=4 [lol])