Journal of Science and Arts Year 14, No. 2(27), pp. 151-158, 2014 ISSN: 1844 – 9581 Mathematics Section ORIGINAL PAPER GROUPLIKE ELEMENTS FOR TRIGONOMETRIC AND HYPERBOLIC COALGEBRAS GEORGIANA VELICU 1 _________________________________________________ Manuscript received: 21.04.2014; Accepted paper: 14.05.2014; Published online: 30.06.2014. Abstract. In this paper we determine all grouplike elements of some some classes of coalgebras over a field k, as well as: trigonometric coalgebras and hyperbolic coalgebras. Also we prove that matrix coalgebra ) , 2 ( k M C doesn’t have any grouplike element. Keywords: grouplike elements, hyperbolic coalgebras, trigonometric coalgebras. 1. GROUPLIKE ELEMENTS FOR THE TRIGONOMETRIC COALGEBRA In the next paragraph we first present the construction of a coalgebra, namely trigonometric coalgebra, starting from the well known trigonometric formulas: . sin sin cos cos ) cos( sin cos cos sin ) sin( y x y x y x y x y x y x ⋅ − ⋅ = + ⋅ + ⋅ = + Because the trigonometric function sin and cos are linear independent, we can consider a two dimension k – linear space T with basis cos} {sin, , which becomes a coalgebra with the comultiplication T T T T → ⊗ ∆ : and the counit k T T → : ε : ) sin( : ) (sin)( y x y x + = + ∆ and sin sin cos cos (cos) ⊗ − ⊗ = ∆ and also: 0 (sin) = ε and 1 (cos) = ε . In general, let’s consider t C be a two dimension k – linear space with basis } , { c s , which becomes a coalgebra with the comultiplication: 1 Valahia University of Targoviste, Faculty of Sciences and Arts, 130024 Targoviste, Romania. E-mail: neacsugeorgiana@yahoo.com .