  In this paper, we reveal some relations between fuzzy logic and quantum
logic, and mainly study the partial residuated implications (PRIs) derived from
partial triangular norms (partial t-norms) and partial residuated lattices
(PRLs), and expand some results in the article "material implication in lattice
effect algebra". Firstly, according to the concept of partial triangular norms
given by Borzooei, we introduce the connection between lattice effect algebra
and partial t-norms, and prove that partial operations in any commutative
quasiresiduated lattice are partial t-norms. Secondly, we give the general form
of partial residuated implications and the concept of partial fuzzy
implications (PFIs), and the condition that partial residuated implication is a
fuzzy implication is given. We also prove that each partial residuated
implication is a partial fuzzy implication. Thirdly, we propose the partial
residuated lattice and study their basic properties, to discuss the
corresponding relationship between PRLs and lattice effect algebras (LEAs), to
further reveal the relationship between LEAs and residuated partial algebras.
In addition, like the definition of partial t-norms, we also propose the
concepts of partial triangular conorms (partial t-conorms) and corresponding
partial co-residuated lattices (PcRLs). Finally, based on partial residuated
lattices, we give the definition of well partial residuated lattices (wPRLs),
study the filter of well partial residuated lattices, and then construct
quotient structure of partial residuated lattices.
