2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168405The notion of monotonic independence, introduced by N. Muraki, is considered in a more general frame, similar to the construction of operator-valued free probability. The paper presents constructions for maps with similar properties to the H and K transforms from the literature, semi inner-product bimodule analogues for the monotone and weakly monotone product of Hilbert spaces, an ad-hoc version of the Central Limit Theorem, an operator-valued arsine distribution as well as a connection to operator-valued conditional freeness.final version, published in PJMOperator AlgebrasFunctional Analysis46L53, 46L08text