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\title{Pseudo-hoops}

\author{George Georgescu and Lauren\c tiu Leu\c stean and Viorel Preoteasa}

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\begin{abstract}
Pseudo-hoops are algebraic structures introduced in \cite{bosbach:1969,bosbach:1970} 
by B. Bosbach under the name of complementary semigroups.
This is a formalization of the paper \cite{georgescu:leustean:preoteasa:2005}.
Following \cite{georgescu:leustean:preoteasa:2005}  we prove some properties 
of pseudo-hoops and we define the basic concepts of filter and normal filter. 
The lattice of normal filters is isomorphic with the lattice of congruences of 
a pseudo-hoop. We also study some important classes of pseudo-hoops. Bounded 
Wajsberg pseudo-hoops are equivalent to pseudo-Wajsberg algebras and bounded 
basic pseudo-hoops are equiv- alent to pseudo-BL algebras. Some examples of 
pseudo-hoops are given in the last section of the formalization.

\end{abstract}

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\section{Overview}
Section 2 introduces some operations and their infix syntax.
Section 3 and 4 introduces some facts about residuated and complemented monoids.
Section 5 introduces the pseudo-hoops and some of their properties. Section 6
introduces filters and normal filters and proves that the lattice of normal 
filters and the lattice of congruences are isomorphic. Following 
\cite{ceterchi:2001}, section 7 introduces pseudo-Waisberg algebras and 
some of their properties. In Section 8 we investigate some classes of 
pseudo-hoops. Finally section 9 presents some examples of pseudo-hoops and
normal filters.

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