Kleene algebras with implication (Q2362883)

The authors study the class of algebras \((H,\wedge,\vee,\to,0,1)\) where the reduct \((H,\wedge,\vee,0,1)\) is a bounded distributive lattice and the operation \(\to\) satisfies natural conditions known in implication algebras. If moreover, the conditions \(a\wedge(a\to b)\leq b\) and \(a\to a = 1\) are satisfied for all \(a,b\) in \(H\), then this algebra is called a \(\mathrm{DLI}^+_1\)-algebra and by \(\mathrm{DLI}^+_1\) the variety of \(\mathrm{DLI}^+_1\)-algebras is denoted. Further, they investigate the category \(\mathrm{KLI}\) whose objects are centered Kleene algebras with involution. In [Trans. Am. Math. Soc. 87, 485--491 (1958; Zbl 0228.06003)], \textit{J. A. Kalman} constructed a functor \(K\) which assigns to every Kleene algebra \(A\) the so-called centered Kleene algebra \(K(A)\), i.e. \(K(A)\) contains an element \(c\) which coincides with its negation. Using this construction for algebras from \(\mathrm{DLI}^+_1\), the authors show the categorical equivalence between \(\mathrm{DLI}^+_1\) and a subcategory of \(\mathrm{KLI}\). Other subcategories of \(\mathrm{KLI}\) (named \(\mathrm{RWH}\), \(\mathrm{SRL}\)) are investigated and the categorical equivalence as well as the corresponding functors and their possible right adjoints are described.