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Journal articles on the topic 'Partial preorder'
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1
Miller, John Boris. "The natural ordering on a strictly real Banach algebra." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 3 (May 1990): 539–56. http://dx.doi.org/10.1017/s0305004100068808.
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AbstractA strictly real unital Banach algebra is one in which every element has real spectrum. An antilattice partial order and its associated open-interval topology are defined on by taking as positive cone the principal component of the maximal group of the algebra, and their properties are studied. The topology coincides with the semimetric topology of the spectral radius, which is a seminorm, making into a locally convex partially ordered topological algebra with continuous inversion and normal cone. Every positive element has a unique positive square root, and logarithm, and these functions and exp are continuous and monotone. The cone has the same closure in both open-interval and normal topologies, the closure being a wedge for an associated preorder on , this preorder coinciding with the preorder of Kelley and Vaught. The partial order and the associated preorder have the same dual cone. There is a modified Gelfand representation for (possibly non-commutative) , and a boundary integral representation for functionals in the order dual. The compact convex subset in the dual is metrizable if the cone of the partial order in is separable.
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Zoethout, Jetze. "On (co)products of partial combinatory algebras, with an application to pushouts of realizability toposes." Mathematical Structures in Computer Science 31, no. 2 (February 2021): 214–33. http://dx.doi.org/10.1017/s096012952100013x.
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AbstractWe consider two preorder-enriched categories of ordered partial combinatory algebras: OPCA, where the arrows are functional (i.e., projective) morphisms, and OPCA†, where the arrows are applicative morphisms. We show that OPCA has small products and finite biproducts, and that OPCA† has finite coproducts, all in a suitable 2-categorical sense. On the other hand, OPCA† lacks all nontrivial binary products. We deduce from this that the pushout, over Set, of two nontrivial realizability toposes is never a realizability topos. In contrast, we show that nontrivial subtoposes of realizability toposes are closed under pushouts over Set.
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French, Zack, and James B. Hart. "The predicate completion of a partial information system." Mathematica Slovaca 68, no. 2 (April 25, 2018): 253–70. http://dx.doi.org/10.1515/ms-2017-0098.
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Abstract Originally, partial information systems were introduced as a means of providing a representation of the Smyth powerdomain in terms of order convex substructures of an information-based structure. For every partial information system 𝕊, there is a new partial information system that is natrually induced by the principal lowersets of the consistency predicate for 𝕊. In this paper, we show that this new system serves as a completion of the parent system 𝕊 in two ways. First, we demonstrate that the induced system relates to the parent system 𝕊 in much the same way as the ideal completion of the consistency predicate for 𝕊 relates to the consistency predicate itself. Second, we explore the relationship between this induced system and the notion of D-completions for posets. In particular, we show that this induced system has a “semi-universal” property in the category of partial information systems coupled with the preorder analog of Scott-continuous maps that is induced by the universal property of the D-completion of the principal lowersets of the consistency predicate for the parent system 𝕊.
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Roldán, López, and Naseer Shahzad. "From graphical metric spaces to fixed point theory in binary related distance spaces." Filomat 31, no. 11 (2017): 3209–31. http://dx.doi.org/10.2298/fil1711209r.
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Very recently, many fixed point results have been introduced in the setting of graphical metric spaces. Due to their intimate links, such works also deal with metric spaces endowed with partial orders. As the reachability relationship in any directed graph (containing all cycles) is a reflexive transitive relation (that is, a preorder), but it is not necessarily a partial order, results on graphical metric spaces are independent from statements on ordered metric spaces. The main aim of this paper is to show that fixed point theorems in the setting of graphical metric spaces can be directly deduced from their corresponding results on measurable spaces endowed with a binary relation. Finally, we also describe the main advantages of involving this last class of spaces.
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Ali, Mabruka, Adem Kiliçman, and Azadeh Zahedi Khameneh. "Application of Induced Preorderings in Score Function-Based Method for Solving Decision-Making with Interval-Valued Fuzzy Soft Information." Mathematics 9, no. 13 (July 4, 2021): 1575. http://dx.doi.org/10.3390/math9131575.
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Ranking interval-valued fuzzy soft sets is an increasingly important research issue in decision making, and provides support for decision makers in order to select the optimal alternative under an uncertain environment. Currently, there are three interval-valued fuzzy soft set-based decision-making algorithms in the literature. However, these algorithms are not able to overcome the issue of comparable alternatives and, in fact, might be ignored due to the lack of a comprehensive priority approach. In order to provide a partial solution to this problem, we present a group decision-making solution which is based on a preference relationship of interval-valued fuzzy soft information. Further, corresponding to each parameter, two crisp topological spaces, namely, lower topology and upper topology, are introduced based on the interval-valued fuzzy soft topology. Then, using the preorder relation on a topological space, a score function-based ranking system is also defined to design an adjustable multi-steps algorithm. Finally, some illustrative examples are given to compare the effectiveness of the present approach with some existing methods.
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Burdick, Bruce S. "On the use of partial orders in uniform spaces." Applied General Topology 4, no. 1 (April 1, 2003): 99. http://dx.doi.org/10.4995/agt.2003.2013.
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<p>We investigate the use of nets indexed by preorders in uniform spaces. Nine different Cauchy conditions and four different convergence conditions yield 36 completeness properties, each of which turns out to be equivalent to a known form of completeness. We also use these preordered nets to characterize the functors θ, λ, and v, which are associated with these completeness properties. In the case of λ we give an example to show that the analogous characterization with predirected nets does not work.</p>
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Babicová, Dušana. "Probability Integral as a Linearization." Tatra Mountains Mathematical Publications 72, no. 1 (December 1, 2018): 1–15. http://dx.doi.org/10.2478/tmmp-2018-0017.
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Abstract In fuzzified probability theory, a classical probability space (Ω, A, p) is replaced by a generalized probability space (Ω, ℳ(A), ∫(.) dp), where ℳ(A) is the set of all measurable functions into [0,1] and ∫(.)dp is the probability integral with respect to p. Our paper is devoted to the transition from p to ∫(.) dp. The transition is supported by the following categorical argument: there is a minimal category and its epireflective subcategory such that A and ℳ(A) are objects, probability measures and probability integrals are morphisms, ℳ(A) is the epireflection of A, ∫(.) dp is the corresponding unique extension of p, and ℳ(A) carries the initial structure with respect to probability integrals. We discuss reasons why the fuzzy random events are modeled by ℳ(A) equipped with pointwise partial order, pointwise Łukasiewicz operations (logic) and pointwise sequential convergence. Each probability measure induces on classical random events an additive linear preorder which helps making decisions. We show that probability integrals can be characterized as the additive linearizations on fuzzy random events, i.e., sequentially continuous maps, preserving order, top and bottom elements.
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Petrović, Marijana, Nataša Bojković, Mladen Stamenković, and Ivan Anić. "A Sensitivity Analysis of ELECTRE Based Stepwise Benchmarking for Policy: the Case of EU Digital Agenda." Management:Journal of Sustainable Business and Management Solutions in Emerging Economies 23, no. 2 (March 6, 2018): 57. http://dx.doi.org/10.7595/management.fon.2018.0003.
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Research question: This paper introduces a new parameter for evaluating the sensitivity of development paths in ELECTRE (ELimination Et Choix Traduisant la Realité) based stepwise benchmarking model. Motivation: The aim of the research was to devise a 'path sensitivity coefficient' which measures the sensitivity of a development path depending on parameters set by decision-maker (DM). The need for such measure comes from the fact that the hierarchical pre-order of benchmarking units are affected by the DM's subjective decisions on the threshold for declaring outranking relations among them. This consequently may disrupt the sequence of intermediate benchmarks - transitional targets to be followed on the route to the ultimate policy goals. The proposed approach is an extension of a stepwise benchmarking procedure devised by Petrovic et al. (2014) which evaluates development paths based on their gradualism. Idea: In this paper, we propose to characterize development paths by considering both gradualism and sensitivity to outranking thresholds. The standpoint of our research is that in order to make a reliable final choice the DM should be aware that the most gradual path is not necessary the most stable one. Tools: The proposed 'path sensitivity coefficient' combines four level of uncertainty coming from two threshold values set by DM. The coefficient has a relative nature, it points to the “distance” from the theoretically ideal solution - a path fully indifferent to threshold values. An approach to merge coefficients associated with gradualism and stability is also introduced. Data: To illustrate the proposed approach we applied it in the field of digital transformation of EU countries. We exploited indicators and data from Digital Agenda Scoreboard regarding specific policy goals to be achieved until 2020. Findings: The findings imply that the decision on the most gradual development path should be reconsidered in regards to the influence of model parameters. The final choice depends whether DM’s prefers the smoothness of the path or the overall stability in terms of underlying partial preorder. Contribution: This work contributes to the existing ELECTRE-based stepwise benchmarking by reducing the uncertainty coming from subjective parameterization and allows the DM to be more confident in the final path selection.
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Carosi, L., and A. Zaffaroni. "On the existence of maximal elements for partial preorders." Journal of Information and Optimization Sciences 20, no. 2 (May 1999): 271–86. http://dx.doi.org/10.1080/02522667.1999.10699417.
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Xu, Xiaozhan, Jean-Marc Martel, and Bernard F. Lamond. "A multiple criteria ranking procedure based on distance between partial preorders." European Journal of Operational Research 133, no. 1 (August 2001): 69–80. http://dx.doi.org/10.1016/s0377-2217(00)00184-3.
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Ésik, Zoltán. "Axiomatizing the subsumption and subword preorders on finite and infinite partial words." Theoretical Computer Science 273, no. 1-2 (February 2002): 225–48. http://dx.doi.org/10.1016/s0304-3975(00)00442-4.
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Aravanis, Theofanis, Pavlos Peppas, and Mary-Anne Williams. "Full Characterization of Parikh's Relevance-Sensitive Axiom for Belief Revision." Journal of Artificial Intelligence Research 66 (November 26, 2019): 765–92. http://dx.doi.org/10.1613/jair.1.11838.
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In this article, the epistemic-entrenchment and partial-meet characterizations of Parikh's relevance-sensitive axiom for belief revision, known as axiom (P), are provided. In short, axiom (P) states that, if a belief set $K$ can be divided into two disjoint compartments, and the new information $\varphi$ relates only to the first compartment, then the revision of $K$ by $\varphi$ should not affect the second compartment. Accordingly, we identify the subclass of epistemic-entrenchment and that of selection-function preorders, inducing AGM revision functions that satisfy axiom (P). Hence, together with the faithful-preorders characterization of (P) that has already been provided, Parikh's axiom is fully characterized in terms of all popular constructive models of Belief Revision. Since the notions of relevance and local change are inherent in almost all intellectual activity, the completion of the constructive view of (P) has a significant impact on many theoretical, as well as applied, domains of Artificial Intelligence.
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Chvalina, Jan, and Šárka Hošková-Mayerová. "On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 1 (December 10, 2014): 85–103. http://dx.doi.org/10.2478/auom-2014-0008.
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AbstractThe contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.
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Berzig, Maher, Erdal Karapınar, and Antonio-Francisco Roldán-López-de-Hierro. "Discussion on Generalized-(αψ,βφ)-Contractive Mappings via Generalized Altering Distance Function and Related Fixed Point Theorems." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/259768.
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We extend the notion of (αψ,βφ)-contractive mapping, a very recent concept by Berzig and Karapinar. This allows us to consider contractive conditions that generalize a wide range of nonexpansive mappings in the setting of metric spaces provided with binary relations that are not necessarily neither partial orders nor preorders. Thus, using this kind of contractive mappings, we show some related fixed point theorems that improve some well known recent results and can be applied in a variety of contexts.
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Urazaeva, T. A. "Relational Theory of Risk and Its Applications to Game Theory Problems of Non-Numerical Economics." Statistics and Economics 18, no. 2 (May 12, 2021): 12–21. http://dx.doi.org/10.21686/2500-3925-2021-2-12-21.
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The purpose of the work is to study the foundations of a general risk theory. To form a single formal concept of risk, a number of definitions of the term “risk” found in the literature have been analyzed. There is a certain redundancy in the number of entities involved in the definition of this term. The necessary attributes of the genesis of this concept have been identified. Based on the analysis, using the instrumentation of modern algebra, a new formal, mathematically strict definition of risk is built. In fact, the paper proposes a new relational theory of risk, attracting only two entities to define the concept of “risk”: set and order of preference, inducing on this set the minimum structure of a semilattice or family of semilattices. The paper also describes an approach for studying in theoretical-risk setting problems in which there is no risk, in the traditional sense, in which the preference relation does not induce a semilattice and/or is a preorder. It is shown that in this case, when identifying a suitable equivalence relation on a set of outcomes, the problem can be reduced to a classical theoretical-risk setting. The second part of the paper contains an example of the direct use of a new relational risk theory in the study of nonnumerical economies. The problem of analysis of the situation of confrontation between two technologically unequal countries in game-theoretical staging is considered. We are talking about a grandiose space program - the organization of manned flights to Mars. The scales of the object of confrontation, the impossibility of quantitative assessments of the consequences of the implementation of scenarios of this project, make any quantitative assessments impossible at the stage of preliminary analysis. Therefore, only expert estimates of preferences at multiple outcomes can be used as initial data for confrontation analysis. Under these conditions, the emergence of certain risks of the project implementation for both players was demonstrated. During the analysis of the example illustrating the application of the new relational risk theory, a number of optimality principles considered in game theory were extended to the case when only partial orders are given on a set of game outcomes. As the methodological basis of the research we used the achievements of modern algebra, in particular the theory of relational systems, as well as the concepts and methods of game theory, such as representing the game in normal form, selecting dominant and eliminating dominant strategies, choosing solutions from many cautious strategies, as well as from the set of Nash Equilibria. The main result of the paper is the substantiation of relational risk theory, the formation of its conceptual base, the demonstration of the constructive nature of the theory on the example of solving a specific problem of risk analysis in an economic system described in terms of non-numerical characteristics. The material presented in the article is of interest to researchers in the field of risk theory and game theory, as well as to practitioners engaged in socio-economic and political forecasting in conditions of lack of information.
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Ivanov, Ievgen, Mykola Nikitchenko, and Uri Abraham. "Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm." Formalized Mathematics 23, no. 4 (December 1, 2015): 325–31. http://dx.doi.org/10.1515/forma-2015-0026.
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Summary Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. Analysis of the literature [1], [7], [12], [9], [13], [14], [15] shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems. In the paper [2] we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson’s algorithm [11]. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in [8]. Using Mizar [3], in this paper we give a formal proof of the mutual exclusion property of the Peterson’s algorithm in Mizar on the basis of the event-based approach proposed in [2]. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation ⩽ on the set of events which turns it into a linear preorder (events are considered simultaneous, if e1 ⩽ e2 and e2 ⩽ e1), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function “val of” on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) [2]. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in [2]). After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson’s algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events e1 and e2 in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson’s algorithm (setting the flag of the current process, writing the identifier of the opposite process to the “turn” shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the “turn” variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the “turn” variable, then either the events e1 and e2 coincide, or they are not simultaneous (mutual exclusion property).
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Vrana, Péter. "A Generalization of Strassen’s Theorem on Preordered Semirings." Order, July 1, 2021. http://dx.doi.org/10.1007/s11083-021-09570-7.
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AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.
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Aceto, Luca, Willem Jan Fokkink, and Anna Ingólfsdóttir. "Ready To Preorder: Get Your BCCSP Axiomatization for Free!" BRICS Report Series 14, no. 3 (February 11, 2007). http://dx.doi.org/10.7146/brics.v14i3.21926.
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This paper contributes to the study of the equational theory of the semantics in van Glabbeek's linear time - branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respectively, ground-complete) equational axiomatization of a behavioral congruence lying between ready simulation equivalence and partial traces equivalence from a complete (respectively, ground-complete) inequational axiomatization of its underlying precongruence--that is, of the precongruence whose kernel is the equivalence. The algorithm preserves finiteness of the axiomatization when the set of actions is finite. It follows that each equivalence in the spectrum whose discriminating power lies in between that of ready simulation and partial traces equivalence is finitely axiomatizable over the language BCCSP if so is its defining preorder.
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Broniek, Przemyslaw. "Solving equations over small unary algebras." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AF,..., Proceedings (January 1, 2005). http://dx.doi.org/10.46298/dmtcs.3474.
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International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.
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Dolinka, Igor, Ivana Đurđev, and James East. "Sandwich semigroups in diagram categories." International Journal of Algebra and Computation, August 13, 2021, 1–66. http://dx.doi.org/10.1142/s021819672150048x.
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This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley–Lieb categories. If [Formula: see text] denotes any of these categories, and if [Formula: see text] is a fixed morphism, then an associative operation [Formula: see text] may be defined on [Formula: see text] by [Formula: see text]. The resulting semigroup [Formula: see text] is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green’s relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green’s classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.