Generalized Bosbach states: part I

Bounded pseudo-hoops with internal states

“…It was also proved in [8] It is sufficient to assume that just one of the arrows →, is preserved, the preservance of the second one is implied by Proposition 3.2 (12). It follows that: If σ preserves one of the arrows →, , then by Proposition 3.2(12) the second one is also preserved.…”

“…which are equivalent according to Proposition 3.2 (12). (3) Since A is idempotent, x y = x ∧ y for all x, y ∈ A and we get:…”

“…Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure of [0, 1], for the case of the residuated lattices the notion of a state was generalized as a function with values in a residuated lattice ( [12], [13]). Recently, this concept was extended to the case of pseudo-BCK algebras and pseudo-hoops ( [14]).…”