Calculi over minimal logic and nonembeddability of algebras

Algebraic semantics of the minimal logic $\mathrm{J}$ is constructed by using Johansson algebras ($\mathrm{J}$-algebras). In this paper the description of Heyting algebras in terms of nonembeddability of $\mathrm{J}$-algebras was found. As a corollary the characterization of superintuitionistic, wellcomposed and some other calculi in the class of various calculi over $\mathrm{J}$ was found.
The central role is played by a special $\mathrm{J}$-algebra $M_{0,\omega}$, constructed and described in this paper.