On canonicity of poset expansions

In this paper, we discuss the canonical extension of poset expansions. To obtain canonicity results on poset expansions, we study Ghilardi and Meloni’s canonicity methodology for Heyting algebras with unary modalities, raise the problem of extending the technique to poset expansions, and give a possible solution for the problem. Finally, we obtain a syntactic account of canonical inequalities on poset expansions consisting of constants, e_^{\epsilon_{\bot}} -additive operations, e^T{\epsilon^{\top}} -multiplicative operations, diamond, box, and strict adjoint pairs, and bounded poset expansions consisting of constants, e{\epsilon} -join preserving operations, e{\epsilon} -meet preserving operations, e{\epsilon} -additive operations, e{\epsilon} -multiplicative operations and adjoint pairs, which are more restricted than the case of lattice expansions, but can still account for Sahlqvist-like canonicity results.