On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). This revised version was published online in June 2006 with corrections to the Cover Date.     

Connections Between BCK-algebras and Difference Posetse

We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.     

States on Pseudo MV-Algebras

Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.     

Conscious and unconscious sensory inflows allow effective control of the functions of the human brain and heart at the initial ageing stage

The authors of the present article based their assumption on the concept that the sensory systems are the "windows to the brain" through which various functions of the human organism can be controlled. Comprehension of the fundamental mechanisms of the optimization of the sensory systems, brain, and cardiac functions has increased based on the prolonged sensory flows using conscious and unconscious aromatherapy and multimodal sensory activation. Sensory flow evoked stable systemic responses, including adaptive alteration of psycho-emotional state, attention, memory, sensorimotor reactions, intersensory interaction, visual information processing, statokinetic stability, and autonomic heart rhythm control. The efficacy and expediency of the use of sensory flow for non-medicinal correction of vital functions of the human organism at the initial stages of ageing was revealed.