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Let $\precsim$ be a reflexive binary relation on a topological space $(X, \tau )$. A pair $(u,v)$ of continuous real-valued functions on $(X, \tau )$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$, [$(x \precsim y \Leftrightarrow u(x) \leq v(y))$]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions $u$ and $v$ nor the topological space $(X,\tau )$ are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation $\precsim$, called {\em weak continuity}. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.

In this paper we focus our efforts on studying how a preorder and topology can be made compatible. Thus we provide a characterization of those that are continuous-compatible. Such a characterization states that such topologies must be finer than the so-called upper topology induced by the preorder and, thus, it clarifies which topology is the smallest one among those that make the preorder continuous. Moreover, we provide sufficient conditions that allows us to discard in an easy way the continuity of a preference. In the light of the obtained results, we provide possibility counterparts of the a few celebrate impossibility theorems for continuous social social intergenerational preferences due to P. Diamond, L.G. Svensson and T. Sakai. Furthermore, we suggest quasi-pseudo-metrics as appropriate quantitative tool for reconciling topology and social intergenerational preferences. Thus, we develop a metric type method which is able to guarantee possibility counterparts of the aforesaid impossibility theorems and, in addition, it is able to give numerical quantifications of the improvement of welfare. We also show that our method makes always the intergenerational preferences semi-continuous multi-utility representables in the sense of \"{O}zg\"{u} Evern and Efe O. Ok. Finally, in order to keep close to the classical way of measuring in the literature, a refinement of the previous method is presented in such a way that metrics are involved.

This paper presents a new and original first approach to agreement situations as well as to regulations constructions. This is made by giving a mathematical formalization to the set of all possible deals or regulations, such that then, the proximity between them is defined by means of a premetric. Thanks to this mathematical structure that tries to capture the problematic of agreements and regulation modifications, now some questions related to game theory or law are reduced to mathematical optimization problems.

The celebrated Kleene fixed point theorem is crucial in the mathematical modelling of recursive specifications in Denotational Semantics. In this paper we discuss whether the hypothesis of the aforementioned result can be weakened. An affirmative answer to the aforesaid inquiry is provided so that a characterization of those properties that a self-mapping must satisfy in order to guarantee that its set of fixed points is non-empty when no notion of completeness are assumed to be satisfied by the partially ordered set. Moreover, the case in which the partially ordered set is coming from a quasi-metric space is treated in depth. Finally, an application of the exposed theory is obtained. Concretely, a mathematical method to discuss the asymptotic complexity of those algorithms whose running time of computing fulfills a recurrence equation is presented. Moreover, the aforesaid method retrieves the fixed point based methods that appear in the literature for asymptotic complexity analysis of algorithms. However, our new method improves the aforesaid methods because it imposes fewer requirements than those that have been assumed in the literature and, in addition, it allows to state simultaneously upper and lower asymptotic bounds for the running time computing.

In the present paper we study necessary and sufficient conditions for the existence of a semicontinuous and finite Richter-Peleg multi-utility for a preorder. It is well know that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition does not guarantee the existence of a semicontinuous representation. We search for finer topologies which are necessary for semicontinuity, as well as that they could guarantee the existence of a semicontinuous representation. As a result, we prove that Scott topology (that refines the Upper one) must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is shown, the existence of this representation cannot be guaranteed.

In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability,...) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.