  In this paper we study the total positivity of almost-Riordan arrays
$(d(t)|\, g(t), f(t))$ and establish its necessary conditions and sufficient
conditions, particularly, for some well used formal power series $d(t)$. We
present a semidirect product of an almost-array and use it to transfer a total
positivity problem for an almost-Riordan array to the total positivity problem
for a quasi-Riordan array. We find the sequence characterization of total
positivity of the almost-Riordan arrays. The production matrix $J$ of an
almost-Riordan array $(d|\, g,f)$ is presented so that $J$ is totally positive
implies the total positivity of both the almost-Riordan array $(d|\, g,f)$ and
the Riordan array $(g,f)$. We also present a counterexample to illustrate that
this sufficient condition is not necessary. If the production matrix $J$ is
tridiagonal, then the expressions of its principal minors are given. By using
expressions, we find a sufficient and necessary condition of the total
positivity of almost-Riordan arrays with tridiagonal production matrices. A
numerous examples are given to demonstrate our results.
