In this paper, we first study the non-positive decreasing and inverse co-radiant FUNCTIONS defined on a REAL locally convex topological vector space X. Next, we characterize non-positive increasing, co-radiant and quasi-concave FUNCTIONS over X. In fact, we examine abstract concavity, upper support set and superdifferential of this class of FUNCTIONS by applying a type of duality. Finally, we present abstract concavity of extended REAL VALUED increasing, co-radiant and quasi-concave FUNCTIONS.

In this paper, for eachlattice-VALUED map A®L with some properties, a RING representation A®RL is constructed. This representation is denoted byc which is an f-RING homomorphism and a Q-linear map, where its indexc, mentions to a lattice-VALUED map. We use the notation a dpqa= (a − p)+^ (q − a)+, where p, q Î Q and a Î A, that is nominated asinterval projection. To get a well-defined f-RING homomorphism tc, we need such concepts asbounded, continuous, and Q-compatible for c, which are defined and some related results are investigated. On the contrary, we present a cozero lattice-VALUED map c f: A®L for each f-RING homomorphism f: A®RL. It is proved that crc=cr and tcf=f, which they make a kind of correspondence relation between RING representations A®RL and the lattice-VALUED maps A®L, where the mapping cr: A®L is called a REALizationof c. It is shown that tcr=tc and crr=cr.

We characterize compact composition operators on REAL Banach spaces of complex-VALUED bounded Lipschitz FUNCTIONS on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.

Let L be a completely regular frame and RL be the RING of REAL-VALUED continuous FUNCTIONS on L. We consider the set R1L = {φ ∈ RL: ↑ φ ( − 1 n, 1 n ) is a compact frame for any n ∈ N}. Suppose that C1(X) is the family of all FUNCTIONS f ∈ C(X) for which the set {x ∈ X: |f(x)| ≥ 1 n } is compact, for every n ∈ N. Kohls has shown that C1(X) is precisely the intersection of all the free maximal ideals of C (X). The aim of this paper is to extend this result to the REAL continuous FUNCTIONS on a frame and hence we show that R1L is precisely the intersection of all the free maximal ideals of R L. This result is used to characterize the maximal ideals in R1L.

It is well known that the sum of two z-ideals in C(X) is either C(X) or a z-ideal. The main aim of this paper is to study the sum of strongly z-ideals in RL, the RING of REAL-VALUED continuous FUNCTIONS on a frame L. For every ideal I in RL, we introduce the biggest strongly z-ideal included in I and the smallest strongly z-ideal containing I, denoted by Isz and Isz, respectively. We study some properties of Isz and Isz: Also, it is observed that the sum of any family of minimal prime ideals in the RING RL is either RL or a prime strongly z-ideal in RL. In particular, we show that the sum of two prime ideals in RL which are not chains is a prime strongly z-ideal.

In this paper, we dene the classes F (p, q, s) of quaternion-VALUED FUNCTIONS, then we characterize quaternion Bloch FUNCTIONS by quaternion F (p, q, s) FUNCTIONS in the unit ball of R3. Further, some important basic properties of these FUNCTIONS are also considered.