In this paper, we define the concept of nucleus map on equality algebras and study related results. Then, using this concept and upsets, a topology on equality algebras is constructed and it is shown that the equality algebra with this topology becomes a topological space. In addition, some properties of topological space such as compactness and connectedness are investigated. Moreover, we study the continuity of all operations with respect to the topology on equality algebras. Finally, the relations between the two topologies in quotient equality algebras are revealed.
In this note, for a certain class of regular continuous frames, we establish conditions that are equivalent to saying that the Freudenthal compactification and the smallest compactification are indistinguishable; in turn, this expands the list of conditions under which the smallest compactification is perfect, which is available in the literature. We define a new class of morphisms between frames, called F-maps, and provide a proof demonstrating that the category of compact regular frames and F-maps forms a coreflective full subcategory of the category of rim-compact frames and F-maps. This coreflection is evidenced by the join map associated with the Freudenthal compactification. Accordingly, this provides an affirmative answer to the question by Herrlich, which inquired whether the Freudenthal compactification can be regarded as a reflection with "sensible" maps.
For a frame $L$, $\mathcal{R}^+(L)$ denotes the nonnegative real valued continuous functions on $L$. We define the concept of $z$-ideals in this semiring and give a characterization of its $z$-ideals in terms of cozero elements of $L$. Also, we show that there is a one-one correspondence between $z$-ideals and $z$-congruences on a ring $\mathcal{R}(L)$ and a semiring $\mathcal{R}^+(L)$. We establish a relationship between $z$-congruence relation on $\mathcal{R}(L)$ and $z$-congruence relation on $\mathcal{R}^+(L)$. A new characterization of $P$-frames is given via $z$-congruences on $\mathcal{R}^+(L)$. Also, we show that there is a bijection between the minimal prime ideals of $\mathcal{R}(L)$ and coz-ultrafilter on $L$.
In this paper we generalize fibrations by $\mathcal{H}$-fibrations, the maps which homotopically lift homotopies. We replace the equalities in the definition of covering homotopy property with the homotopy relation so that we can first get an expression of the concept of covering homotopy property in the homotopy category. After introducing $\mathcal{H}$-fibrations, we will have a homotopy expression of some concepts related to fibration, such as path lifting, lifting function and unique path lifting property, to generalize some results in fibration. In particular, we show that an $\mathcal{H}$-fibration has homotopical path lifting property and also prove that a map is an $\mathcal{H}$-fibration if and only if it has a homotopical lifting function.
In [8] Valdis Laan introduced Condition (PW P). Golchin and Mohammadzadeh in [3] introduced Condition (PW P_E), such that Condition (PW P) implies it but the converse is not true in general. In this paper at first we introduce a generalization of Condition (PW P_E), called Condition (PW P_S). Then will give some general properties and a characterization of monoids for which all right acts satisfy this condition. Also, we give a characterization of monoids, by comparing this property of their acts with some others. Finally, we will give a characterization of monoid S, for which S^{I}_{S}, for any non-empty set I and S^{S \times S}_{S}, satisfy Condition(PW P_S).
