Commutative extended BCK-algebras

We introduce an extended cone algebra, which generalises a Bosbach’s cone algebra within the framework of extended BCK-algebras and show that every such an algebra is a direct product of an ℓ-group and a cone algebra of Bosbach.

Deductive systems of a cone algebra — I: Semi-ℓg-cones

A semi-ℓg-cone is an algebra (C;*,:,·) of type (2, 2, 2) satisfying the equations (a*a)*b = b = b: (a: a); a*(b: c) = (a*b): c; a: (b*a) = (b: a)*b and (ab) *c = b* (a * c). An ℓ-group cone is a semi-ℓg-cone and a bounded semi-ℓg-cone is term equivalent to a pseudo MV-algebra. Also, a subset A of a semi-_g-cone C is an ideal of C if and only if it is a deductive system of its reduct (C;*,:).

Deductive systems of a cone algebra — II: Isomorphism theorem

We prove that there is an isomorphism φ of the lattice of deductive systems of a cone algebra onto the lattice of convex ℓ-subgroups of a lattice ordered group (determined by the cone algebra) such that for any deductive system A of the cone algebra, A is respectively a prime, normal or polar if and only if φ(A) is a prime convex ℓ-subgroup, ℓ-ideal or polar subgroup of the ℓ-group, thus generalizing and extending the result of Rachůnek that the lattice of ideals of a pseudo MV-algebra is isomorphic to the lattice of convex ℓ-subgroups of a unital lattice ordered group.