Axiomatizability of hereditary classes of structures of finite and infinite languages and decidability of their universal theories

In the paper hereditary classes of $\mathrm{L}$-structures are studied with language of the form $\mathrm{L}={\mathrm{L}}_{\mathrm{f}\mathrm{i}\mathrm{n}}\cup {\mathrm{L}}_{\mathrm{\infty }}$, where ${\mathrm{L}}_{\mathrm{f}\mathrm{i}\mathrm{n}}=⟨{\mathrm{R}}_1,{\mathrm{R}}_2,\dots ,{\mathrm{R}}_{\mathrm{m}},=⟩$ and ${\mathrm{L}}_{\mathrm{\infty }}=⟨R_{m+1},R_{m+2},\dots ⟩$, and also in ${\mathrm{L}}_{\mathrm{\infty }}$ the number of predicates of each arity is finite, all predicates are ordered in ascending of their arities and satisfying the property of non-repetition of elements. A class of $\mathrm{L}$-structures is called hereditary if it is closed under substructures. It is proved that class of $\mathrm{L}$-structures is hereditary if and only if it can be defined in terms of forbidden substructures. A class of $\mathrm{L}$-structures is called universal axiomatizable if there is a set $Z$ of universal $\mathrm{L}$-sentences such that the class consists of all structures that satisfy $Z$. The problems of universal axiomatizability of hereditary classes of $\mathrm{L}$-structures are considered in the paper. It is shown that hereditary class of $\mathrm{L}$-structures is universal axiomatizable if and only if it can be defined in terms of finite forbidden substructures. It is proved that the universal theory of any axiomatizable hereditary class of $\mathrm{L}$-structures with recursive set of minimal forbidden substructures is decidable.