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directoid; relatively pseudocomplemented directoid; congruence distributivity; $3$-permutability; residuated structure; adjointness property; variety
The concept of a relatively pseudocomplemented directoid was introduced recently by the first author. It was shown that the class of relatively pseudocomplemented directoids forms a variety whose axiom system contains seven identities. The aim of this paper is three-fold. First we show that these identities are not independent and their independent subset is presented. Second, we modify the adjointness property known for relatively pseudocomplemented semilattices in the way which is suitable for relatively pseudocomplemented directoids. Hence, they can also be considered as residuated structures in a rather modified version. We also get two important congruence properties, namely congruence distributivity and $3$-permutability valid in the variety $\mathcal {V}$ of relatively pseudocomplemented directoids. Then we show some basic results connected with subdirect irreducibility in $\mathcal {V}$. Finally, we show another way how to introduce pseudocomplementation on directoids via relative pseudocomplementation.
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