  Consider a partial order on $\{0,1\}^{\mathbb Z}: x\leq y$ when $x_i\leq y_i$
for all $i\in\mathbb{Z}$. A subshift $X\subset\{0,1\}^{\mathbb{Z}}$ is
hereditary if together with any $x\in \{0,1\}^{\mathbb Z}$ it contains all
$y\leq x$. Heuristically speaking, a hereditary subshift contains all the
elements between maximal elements (with respect to this partial order) and the
element $0^{\mathbb Z}$. In a particular situation when it suffices to take
(the orbit closure of) all the elements between a single maximal element $x$
and the element $0^{\mathbb Z}$, we speak of subordinate subshifts. In this
paper we investigate measure-theoretic properties of such subshifts, with a
special emphasis on thermodynamical formalism. The key notion is a
measure-theoretic counterpart of subordinate subshifts, where the role of a
single maximal element is replaced with a single (maximal with respect to a
certain order) invariant measure on $\{0,1\}^{\mathbb{Z}}$.
  We also introduce and investigate two-sided analogues of the above classes,
we call them {\it sandwich hereditary}, {\it sandwich subordinate} and {\it
sandwich measure-theoretically subordinate} subshifts. Sandwich hereditary
subshifts can be thought of as sets of elements between some pairs of maximal
and minimal elements satisfying certain assumptions. Sandwich subordinate
subshifts occur when it suffices to take (the orbit closure of) all the
elements between a single pair of sequences $(w,x)$, where $w\leq x$. In
sandwich measure-theoretically subordinate subshifts, the role of a pair of
sequences is replaced by a pair (precisely speaking: a joining) of two
invariant measures on $\{0,1\}^{\mathbb{Z}}$.
  The notions and results are motivated by those from the theory of so-called
$\mathscr{B}$-free systems.
