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In the thesis there are constructed new quantizations for pseudo-differential boundary value problems (BVPs) on manifolds with edge. The shape of operators comes from Boutet de Monvel’s calculus which exists on smooth manifolds with boundary. The singular case, here with edge and boundary, is much more complicated. The present approach simplifies the operator-valued symbolic structures by using suitable Mellin quantizations on infinite stretched model cones of wedges with boundary. The Mellin symbols themselves are, modulo smoothing ones, with asymptotics, holomorphic in the complex Mellin covariable. One of the main results is the construction of parametrices of elliptic elements in the corresponding operator algebra, including elliptic edge conditions.

A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. We consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as interassociative semigroups, restrictive bisemigroups, dimonoids, and trioids.
In the lecture notes numerous examples of doppelsemigroups and of strong doppelsemigroups are given. The independence of axioms of a strong doppelsemigroup is established. A free product in the variety of doppelsemigroups is presented. We also construct a free (strong) doppelsemigroup, a free commutative (strong) doppelsemigroup, a free n-nilpotent (strong) doppelsemigroup, a free n-dinilpotent (strong) doppelsemigroup, and a free left n-dinilpotent doppelsemigroup. Moreover, the least commutative congruence, the least n-nilpotent congruence, the least n-dinilpotent congruence on a free (strong) doppelsemigroup and the least left n-dinilpotent congruence on a free doppelsemigroup are characterized.
The book addresses graduate students, post-graduate students, researchers in algebra and interested readers.