A semi-lattice is said to be tree-like when any two of its elements are either orthogonal or comparable. Given an inverse semigroup $$\mathcal{S}$$ whose idempotent tree-like, and such that all tight filters ultra-filters, we present a necessary sufficient condition for contracting which looks closer in spirit the notion actions than found by second named author E. Pardo.

We study heat semigroups generated by self-adjoint Laplace operators on metric graphs characterized by the property that the local scattering matrices associated with each vertex of the graph are independent from the spectral parameter. For such operators we prove a representation for the heat kernel as a sum over all walks with given initial and terminal edges. Using this representation a trac...

In contrast to the semilattice of groups case, an inverse semigroup S which is the union of strongly ^-reflexive inverse subsemigroups need not be strongly £-reflexive. If, however, the union is saturated with respect to the Green's relation <3), and in particular if the union is a disjoint one, then 5 is indeed strongly £-reflexive. This is established by showing that fy -saturated inverse sub...

We analyze convex combinations of non-unital qubit maps that are phase-covariant. In particular, we consider the behavior combine amplitude damping, inverse and pure dephasing. show mixing channels can result in restoring unitality, whereas commutative lead to non-commutativity. For Markovian semigroups, prove classical uncertainties cannot break quantum Markovianity. Moreover, contrary Pauli c...