Co - rotation , co - rotation

Residuated lattices form the algebraic counterpart of substructural logics [4]. Rotation construction for t-norms has been introduced in [9] and has been generalized to arbitrary residuated posets in [7]. The reader is referred also to [13, 6, 5]. The rotationannihilation construction has been introduced in [10, 7]. The rotation construction has proved fundamental in the structural description of perfect and bipartite IMTLalgebras [12], of free nilpotent minimum algebras [1], of free Glivenko algebras [2], and has proved useful in many other mathematical applications. De Morgan dualization can be defined in any algebra which has a binary multiplication ∗◦ and an order-reversing involution ′ on its universe by x ·y = (x∗◦y). However, de Morgan dualization does not fit well to the class of residuated lattices, since de Morgan dual of the monoidal operation of a residuated lattice, in general, is not residuated with respect to the same ordering relation. As an attempt to overcome this, and to describe the structure of a certain class of residuated chains, skew dualization has been introduced in [8]. However, skew-dualization is a notion hard to work with. Two co-rotation constructions and two co-rotation-annihilation constructions (in both cases a disjoint one and a connected one) will be introduced. In some sense these can be considered as ’skew dual’ constructions to the (disjoint and connected) rotation constructions. Just as the rotation(-annihilation) of FLe-algebras results in positive rank involutive FLe-algebras, the co-rotation(-annihilation) of FLe-algebras results in non-positive (zero or negative) rank involutive FLe-algebras; thus providing a wide spectrum of examples for the latter algebra. The fact that co-rotation and co-rotationannihilation constructions are not simply dual of their rotation and rotation-annihilation counterparts is also reflected by the fact that the class of algebras which can be used in them are quite different. Also, a construction, called involutive ordinal sums will be introduced, and its use will be demonstrated in the structural description of a certain class of involutive FLe∗Supported by the SROP-4.2.2.C-11/1/KONV-2012-0005 project.