  In this study, FRW-cosmologies with some matter groups such as
monopole-domain wall, monopole-Chaplygin gas and monopole-strange quark matter
in the scalar theory of gravitation based on Lyra geometry are investigated. We
expand two exact models as static case and time-depended case for each matter
groups in order to solve field equations in the scalar theory. For each matter
groups, the solutions are introduced as the models of expanding universe,
exponentially. Hubble parameters in the case of k=0,-1,1 are obtained for these
models. Furthermore, we realize interesting result which the well-known
relation between scalar theory based on Lyra geometry and Einstein's theory is
an incomplete idea. In opposition to the well accepted idea in the literature,
we suggest that Einstein's theory with no cosmological constant is equivalent
of scalar theory based on Lyra geometry with zero displacement vector,
completely. If the components of displacement vector in the scalar theory are
any constant functions, the scalar theory couldn't correspond to Einstein's
theory, identically. Even if the components of displacement vector in the
scalar theory are the constant, the field equations and their solutions contain
the Einstein's field equations and their solutions, but they are variously more
general than Einstein's theory of gravitation. So, coefficients of constant
displacement vector don't play the role of cosmological constant in keeping
with the Einstein's theory. Finally, the results have been discussed.
