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# Article

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Keywords:
free semigroup; semigroup of matrices
Summary:
Let $$A=\left [ \begin {matrix} 1 & 2 \\ 0 & 1 \end {matrix} \right ],\quad B_{\lambda }=\left [ \begin {matrix} 1 & 0 \\ \lambda & 1 \end {matrix} \right ].$$ We call a complex number $\lambda$ “semigroup free“ if the semigroup generated by $A$ and $B_{\lambda }$ is free and “free” if the group generated by $A$ and $B_{\lambda }$ is free. First families of semigroup free $\lambda$'s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free $\lambda$'s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture.
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