  Tanaka (1997) proved a powerful generalization of Friedman's self-embedding
theorem that states that given a countable nonstandard model
$(\mathcal{M},\mathcal{A})$ of the subsystem $\mathrm{WKL}_{0}$ of second order
arithmetic, and any element $m$ of $\mathcal{M}$, there is a self-embedding $j$
of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such
that $j$ fixes every predecessor of $m$.
  Here we extend Tanaka's work by establishing the following results for a
countable nonstandard model $(\mathcal{M},\mathcal{A})$ of $\mathrm{WKL}_{0} $
and a proper cut $\mathrm{I}$ of $\mathcal{M}$:
  Theorem A. The following conditions are equivalent:
  (a) $\mathrm{I}$ is closed under exponentiation.
  (b) There is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a
proper initial segment of itself such that $I$ is the longest initial segment
of fixed points of $j$.
  Theorem B. The following conditions are equivalent:
  (a) $\mathrm{I}$ is a strong cut of $\mathcal{M} $ and $\mathrm{I}\prec
_{\Sigma _{1}}\mathcal{M}.$
  (b) There is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a
proper initial segment of itself such that $\mathrm{I} $ is the set of all
fixed points of $j$.
