Algebra

In this paper, we extend the notions of states and measures presented in \cite{DvPu} to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of \cite{DvPu} coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. K\"uhr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean $\ell$-group. Additionally, we introduce a Borel state and using results by J. K\"uhr and D. Mundici from \cite{Kumu}, we prove a relationship between de Finetti maps, Bosbach states and Borel states.

The concept of a state MV-algebra was firstly introduced by Flaminio and Montagna in \cite{FlMo0} and \cite{FlMo} as an MV-algebra with internal state as a unary operation. Di Nola and Dvure\v{c}enskij gave a stronger version of a state MV-algebra in \cite{DiDV1}, \cite{DiDV2}. In the present paper we introduce the notion of a state BL-algebra, or more precisely, a BL-algebra with internal state. We present different types of state BL-algebras, like strong state BL-algebras and state-morphism BL-algebras, and we study some classes of state BL-algebras. In addition, we give a sample of important examples of state BL-algebras and present some open problems.

In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen.

The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.

In this paper, the notions of smarandache soft semigroups (SS-semigroups) introduced for the first time. An SS-semigroup
(F, A) is basically a parameterized collection of subsemigroups which has atleast a proper soft subgroup of (F, A) . Some new type of SS-semigroup is also presented here such as smarandache weak commutative semigroup, smarandache weak cyclic semigroup, smarandache hyper subsemigroup etc. Some of their related properties and other notions have been discussed with sufficient amount of examples.

I wrote this paper around a year ago. This paper aims to generalise the Binomial theorem and the Binomial coefficients on Pascal's triangle into a polynomial with n terms

Shortly after writing this paper, I realised what I called the "Polynomial Theorem" already existed and in fact it is called the Multinomial Theorem. However, my work was independent and it was a coincidence that this theorem already existed.