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In the paper hereditary classes of ${\rm L}$-structures are studied withlanguage of the form ${{\rm L} = {\rm L_{fin}} \cup {\rm L_\infty}}$, where${{\rm L_{fin}} = \langle R_1,R_2,\ldots, R_m, = \rangle}$ and ${{\rm L_\infty}= \langle R_{m+1}, R_{m+2}, \ldots \rangle}$, and also in ${\rm L_\infty}$ thenumber of predicates of each arity is finite, all predicates are ordered inascending of their arities and satisfying the property of non-repetition ofelements. A class of ${\rm L}$-structures is called hereditary if it is closedunder substructures. It is proved that class of ${\rm L}$-structures ishereditary if and only if it can be defined in terms of forbiddensubstructures. A class of ${\rm L}$-structures is called universalaxiomatizable if there is a set $Z$ of universal ${\rm L}$-sentences such thatthe class consists of all structures that satisfy $Z$. The problems ofuniversal axiomatizability of hereditary classes of ${\rm L}$-structures areconsidered in the paper. It is shown that hereditary class of ${\rmL}$-structures is universal axiomatizable if and only if it can be defined interms of finite forbidden substructures. It is proved that the universal theoryof any axiomatizable hereditary class of ${\rm L}$-structures with recursiveset of minimal forbidden substructures is decidable.

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