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Let n, m ∈ ℕ, n ≥ 3 and n ≥ m ≥ 2. In this paper, we study some properties of the unrestricted virtual singular braid of braid monoid UVSBmn and group UVSGmn. These properties emerged by studying the analogous properties of the unrestricted virtual singular pseudosymmetric braid of braid monoid and group. These mathematical objects are submonoids and subgroups of the virtual singular braid group VSGn. For many of these quotients, we have obtained reduced presentations.

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]