Pre-anti-flexible family algebras are introduced and used to define describe the notions of Ωc-relative anti-flexible algebras, left right pre-Lie Lie algebras. The notion pre-anti-flexible also characterize significant identities associated these algebraic structures provided. Finally, a generalization Rota-Baxter operators defined on an algebra is it proved that both its provide pre-antiflexi...

Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. These algebras have been extensively studied in recent years, both from algebraic operadic points of view and through numerous applications in numerical analysis, control theory, stochastic differential equations and renormalization. Butcher series are formal power series founded on pre-Lie alg...

The original protocol for implant prosthesis recommends the use of a gold framework for acrylic and ceramic prosthesis. However, due to its high cost, the use of alternative alloys is desired. This study compares the marginal accuracy of pre-made cylinders versus plastic cylinders cast with two different base metal casting alloys. Five samples each of (1) plastic cylinder cast in cobalt-chromiu...

The algebraic approach to formal language and automata theory is a continuation of the earliest traditions in these fields which had sought to represent languages, translations and other computations as expressions (e.g. regular expressions) in suitably-defined algebras; and grammars, automata and transitions as relational and equational systems over these algebras, that have such expressions a...

We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the “alternative” analogue of a dendriform dialgebra or a pre-Lie algebra. The left and right multiplication operators of a pre-alternative algebra give a bimodule structure of the associated alternative al...