  The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is
considered. This presentation was originally introduced by T. Ito and P.
Terwilliger. In this paper, following Terwilliger's recent works the
(nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type
$U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ are
introduced and studied. A presentation for $U_q^{T,+}$ is given in terms of a
K-operator satisfying a Freidel-Maillet type equation and a condition on its
quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel
L-operators are considered, from which an explicit injective homomorphism from
$U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra)
of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of
$U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and its
relation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is also
described using the framework of Freidel-Maillet algebras.
