The class of semi-boolean ℓ -groups was introduced in 1968 by A. Bigard. These are the ℓ -groups G in which the principal convex ℓ -subgroup G(a) generated by any a ∈ G is equal to the polar a^⊥⊥ . Examples include all hyperarchimedean ℓ -groups and all existentially closed abelian ℓ -groups. Ordered by inclusion, the set of convex ℓ -subgroups of a semi-boolean ℓ -group is a Martínez frame (an algebraic frame with FIP in which every element is a d-element). Related are the Yosida ℓ -groups, i.e., the ℓ -groups whose frame of convex ℓ -subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on Martínez frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida ℓ -groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the G+B construction for ℓ -groups, an adaptation of the A+B construction from commutative algebra.

The minimal prime elements of algebraic frames with the finite intersection property (FIP) can be characterized as those prime elements which are the joins of the pseudo-complements of all compact elements not below the said prime. Our goal is to study the class of algebraic frames with this latter property, generalizing the FIP case.

𝔎ℜ𝔢𝔤 is the category of compact regular frames and frame homomorphisms. A class of 𝔎ℜ𝔢𝔤 frames H is a hull class provided that: (i) H is closed under isomorphic copies; (ii) for every F ∈𝔎ℜ𝔢𝔤 there exist an hF ∈H and a morphism h_F such that F h_F≤ hF is essential; (iii) if F ϕ≤ H is essential and H ∈H , then there exists hϕ : hF ⟶ H for which ϕ = hϕ· h_F . This work provides techniques for identifying and generating hull classes in 𝔎ℜ𝔢𝔤 . Moreover, for a compact regular frame F, we introduce and investigate various properties of projectability and disconnectivity of F and prove that for each property, P, the class of 𝔎ℜ𝔢𝔤 -objects that satisfy P is a hull class in 𝔎ℜ𝔢𝔤 . In addition, we provide examples of 𝔎ℜ𝔢𝔤 hull classes that are not characterized by some form of projectability/disconnectivity and examples of classes of 𝔎ℜ𝔢𝔤 -objects that are not hull classes.

A ring is called left fusible if each of its nonzero elements is the sum of a left zero-divisor and a non-left zero-divisor. This paper aims to extend the existing theory of left fusible rings and related notions such as being left unit-fusible, being regular left fusible, and being uniquely left fusible. New examples of these rings are presented. A new notion of left generalized unit-fusible rings is introduced and discussed. The asymmetry of these notions is addressed.

This article studies different topological properties of the space of maximal elements of an M-frame with a unit. We characterize when the space Max(dL) ( is Hausdorff, answering the question posed in [2]. We also characterize other topological properties of Max(dL), ( dL ), namely zero-dimensional, discrete, and clopen base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of d-semiprime frames. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and mining, AI training, and similar technologies.

W* is the category of the title. For G is an element of W*, we have the canonical compact space YG, and Yosida representation G <= C(YG), thus, for g is an element of G, one has the cozero-set coz(g) in YG. The ideals at issue in G include the principal ideals and polars, G(g) and g(perpendicular to perpendicular to), respectively, and the W*-kernels of W*-morphisms from G. The "coincidences of types" include these properties of G: (M) Each G(g) = g(perpendicular to perpendicular to); (Y) Each G(g) is a W*-kernel; (CR) Each g(perpendicular to perpendicular to) is a W*-kernel (iff each coz(g) is regular open). For each of these, we give numerous "rephrasings" and examples, and note that (M) = (Y) boolean AND (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on W* brings topology to bear, and sharpens the view.

An h-local domain is a domain for which each nonzero prime ideal is contained in a unique maximal ideal and each nonzero element has finite character. In his dissertation, A. Omairi generalizes the notion of an h-local domain to rings with zero-divisors by restricting the definition to regular ideals. In this article, we give a characterization of when C(X) the ring of continuous real-valued functions on a space X is an h-local ring. We are left with more questions than answers.

We call a frame Mart & imath;nez if it is an algebraic frame with FIP in which every element is a d-element. The study of Martinez frames and the d-operator in this article has led to a better understanding of the q-nucleus defined in [15]. We generalize the construction of the q-nucleus to arbitrary sets of primes and investigate this operator from a topological perspective on the prime spectrum.

We abstract the notion of fraction-density of f -rings (introduced by Anthony Hager and Jorge Martínez) to algebraic frames. We say an algebraic frame with the finite intersection property on compact elements is fraction-dense if each of its polars is a polar of a compact element. This turns out to be a “conservative” extension of the fraction-density property in the sense that a reduced f -ring is fraction-dense precisely when its frame of radical ideals is fraction-dense. We characterize these frames and study properties of some other types of algebraic frames that arise naturally in the characterizations of the fraction-dense ones.

It was demonstrated in [2] that the Alexandroff duplicate of the Čech-Stone compactification of the naturals is not extremally disconnected. The question was raised as to whether the Alexandroff duplicate of a non-discrete extremally disconnected space can ever be extremally disconnected. We answer this question in the affirmative; an example of van Douwen is significant. In a slightly different direction we also characterize when the Alexandroff duplicate of a space is a P-space as well as when it is an almost P-space.

In [17] pointfree compactness and precompactness are characterized by the convergence and clustering of classical filters, ultrafilters and Cauchy filters. Banaschewski in [2] introduced the notion of general filters as bounded meet semilattice homomorphisms. Together with Hong in [4], [5], [6], [7] they described the notion of convergence of such filters as those that are cover preserving. In this paper, we revisit the notion of general filters in a frame and introduce the concepts of clustering general filters, maximal general filters and general ultrafilters. These concepts have hitherto not been considered for general filters. We use these concepts to characterize, among other things, almost compact frames, Boolean frames and precompact uniform frames.

The space of maximal d-ideals of C(X) is well-known and is widely studied. It is known that the space of maximal d-ideals is homeomorphic to the Z♯(X)-ultrafilters, and this space is the minimal quasi F-cover of a compact Tychonoff space X. In the current article we generalize this concept for M-frames, algebraic frames with the finite intersection property. In particular, we explore various properties of the maximal d-elements of a frame L, and their relation with the ultrafilters of \(\mathfrak {K}L^{\perp }\), the polars of the compact elements of L. On a separate note, we revisit the Lemma on Ultrafilters and establish the correspondence between the minimal prime elements spaces of L with the spaces of ultrafilters of \(\mathfrak {K}L\). Finally, we show that for complemented frames L, Min(L) = Max(dL), a result parallel to the one known for Riesz spaces, W-objects, and topological spaces.

Appointment scheduling plays a key role in improving the performance of a healthcare facility and increasing patient access to health care. However, appointment systems in hospitals may receive requests for services from walk-ins and emergency arrivals in addition to the scheduled arrivals. Emergency arrivals need urgent care, and hence have higher priority to be served over the scheduled arrivals. Such emergency arrivals disrupt the scheduled appointments, and may increase the waiting times of scheduled patients and the overtime of the appointment session. Walk-ins have lower priority to be served. However, it is desirable to serve walk-ins in order to increase the utilization of the server keeping the waiting times of scheduled patients as short as possible. In this research work, appropriate sequencing and appointment rules are identified for a computed tomography (CT) scan facility that experiences walk-in and emergency arrivals apart from regular scheduled arrivals. Several scheduling rules are evaluated under different patterns of walk-ins and emergency arrivals. In addition, the main and interaction effects of scheduling rules and arrival patterns of unscheduled patients are explored. It is found that the scheduling rules are more influential than the arrival patterns of unscheduled patients.

The article introduces a new class of lattice-ordered groups. An -group G is lamron if Min(G)(-1) is a Hausdorff topological space, where Min(G)(-1) is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron -groups are related to -groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron -group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented -groups.

In this article we investigate filters of cozero sets for real-valued continuous functions, called $coz$-filters. Much is known for $z$-ultrafilters and their correspondence with maximal ideals of $C(X)$. Similarly, a correspondence will be established between $coz$-ultrafilters and minimal prime ideals of $C(X)$. We will further notice various properties of $coz$-ultrafilters in relation to $P$-spaces and $F$-spaces. In the last two sections, the collection of $coz$-ultrafilters will be topologized, and then compared to the hull-kernel and the inverse topologies placed on the collection of minimal prime ideals of $C(X)$ and general lattice-ordered groups.

We study the space \({{\mathrm{Max~}}}_d(G)\) of maximal d-subgroups of a lattice-ordered group, paying specific attention to archimedean \(\ell \)-groups with weak order unit. For such an object (G, u), \({{\mathrm{Max~}}}_d(G)\) lays at a level in between the space of minimal prime subgroups and the Yosida space of (G, u). Theorem 5.10 gives the appropriate generalization of a quasi F-space to W-objects which avoids a discussion of o-complete \(\ell \)-groups.

Appointment systems in many service facilities, especially in hospitals, experience walk-ins and emergency arrivals in addition to the scheduled arrivals. While the emergency arrivals need to be served as early as possible with the highest priority, the walk-ins are usually served with the lowest priority and are served on the occurrence of no-shows. This may disrupt the original schedule and affect the performance of the appointment system related to the scheduled patients (their waiting time). In this chapter, an attempt has been made to draw insights about the appropriate appointment scheduling rules for multiple classes of scheduled patients in presence of walk-ins and emergency patients. A number of simple scheduling rules are evaluated under different patterns of walk-ins and emergency arrivals. The analyses are illustrated with an example appointment system of a CT scan department of a hospital.

Appointment systems for scheduling patients to a hospital facility play an important role in controlling and synchronizing the arrival of patients with resource availability thereby reducing the waiting time of patients and increasing the utilization of resources. In this paper, hospital appointment systems with multiple classes of patients are considered where different classes of patients may vary in punctuality, no-show probabilities, mean service times and service time variability. Classification of patients may help in providing important insights for designing an appointment system considering different sequencing schemes and adjustments of inter-appointment times. For illustration, a case study is conducted for appointment scheduling of patients to the Magnetic Resonance Imaging (MRI) scanning machine of the Radiology department of a multi-speciality hospital situated in eastern India. The patient flows in the system are modelled using discrete-event simulation wherein a number of appointment scheduling policies (combinations of different sequencing and appointment rules) are analysed and evaluated. The performance measures of interest include the waiting times of all classes of patients and utilization of the server (MRI machine). The simple sequencing rule of ordering the patient classes in increasing order of mean service time performs best among all the sequencing rules. The individual block variable interval appointment rule, i.e., scheduling a single patient at a time with inter-appointment times adjusted according to the mean service times of patient class performs best.

An algebraic frame L with the finite intersection property (FIP) on compact elements is said to be polarised if every minimal prime element in it is complemented. In this note, we give a necessary and sufficient condition for the inverse topology on the set of minimal prime elements of such a frame to be sober. We also establish some sufficient conditions for sobriety when the polarisation condition is relaxed.