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Stimulus equivalence paradigm has been used to investigate symbolic behavior over the last 50 years. Still, little attention has been directed in the study of equivalence classes maintenance. This bibliographic review, based on the PRISMA Protocol, provides an overview of the effects of different variables on the stability of equivalence classes over time. Searches were conducted on four databases (PsychINFO, Elsevier Scopus, Web of Science, and SciELO) and returned 95 results. Excluding the duplicated papers and applying the inclusion and exclusion criteria resulted in 14 papers. The references of each paper were examined for papers about maintenance of equivalence relations that were not found in the database search and 10 more articles were included. Therefore, 24 papers were selected in this bibliographic review. The papers were categorized considering the variable investigated, that is: class size, training and testing parameters, nature of stimuli, generalized relations, and equivalence based instruction. Future studies should focus on the study of variables that affect the maintenance of equivalence classes, considering the small number of papers in this field and the importance of this knowledge in both experimental and applied settings.

This study aimed to verify the role of three parameters on the formation of equivalence classes between Black faces and a positive symbol, in children who demonstrated negative bias toward Black faces in a pretest. Maintenance was also verified 6 weeks after equivalence tests. Forty-six children (11 Black; 27 girls) who demonstrated racial bias in a pretest were divided into four groups. All groups first learned AB relations (A1 and A2 were, respectively, a positive and a negative symbol, and B were abstract stimuli) and then BC relations (C1 was a Black face and C2 was an abstract stimulus). The Control Group then advanced immediately to equivalence tests (AC, and CA, without differential consequences). For the Mixed Training Group, a block of trials mixing AB and BC relations, with differential consequences, preceded equivalence tests. For the Feedback Reduction Group, equivalence tests were preceded by a trial block mixing AB and BC relations, but with feedback in 50% of trials. The Symmetry Group received symmetry tests after training of each baseline relation. Thirty-three children showed class formation relating Black faces and the positive symbol, and 27 maintained at least one of the equivalence relations after 6 weeks. Average biases toward Black faces were positive in a posttest, for participants who formed equivalence classes, and remained negative for those that did not form classes. The Control Group showed less pronounced bias reduction and maintenance of relations after 6 weeks, suggesting that these outcomes may be affected by training parameters.

“Yield,” the percentage of participants in a group who form a set of equivalence classes, has been used very broadly to identify the effect of different training protocols on class formation and expansion, identify variables that enhance the immediate emergence of these classes, and characterize the differential relatedness of class members. In addition, yield is now being used to document the formation of educationally relevant equivalence classes. To further understand the value of using yield, we considered six possible criticisms of its use to study equivalence classes. Upon analysis, each criticism was supported; instead, each disclosed a nonyield factor that could play a critical role in the measurement of class formation but has not yet been explored experimentally. Finally, yield cannot be replaced with trial-based measures of responding or vice versa; rather, both types of measures are needed to obtain a comprehensive understanding of equivalence class formation.

Thirty college students attempted to form three 3‐node 5‐member equivalence classes under the simultaneous protocol. After concurrent training of AB, BC, CD, and DE relations, all probes used to assess the emergence of symmetrical, transitive, and equivalence relations were presented for two test blocks. When the A‐E stimuli were all shapes, none of 10 participants formed classes. When the A, B, D, and E stimuli were shapes and the C stimuli were meaningful pictures, 8 of 10 participants formed classes. This high yield may reflect the expansion of existing classes that consist of the associates of the meaningful stimuli, rather than the formation of the ABCDE classes, per se. When the A‐E stimuli were shapes and the C stimuli became SDs prior to class formation, 5 out of 10 participants formed classes. Thus, the discriminative functions served by the meaningful stimuli can account for some of the enhancement of class formation produced by the inclusion of a meaningful stimulus as a class member. A sorting task, which provided a secondary measure of class formation, indicated the formation of all three classes when the emergent relations probes indicated the same outcome. In contrast, the sorting test indicated “partial” class formation when the emergent relations test indicated no class formation. Finally, the effects of nodal distance on the relatedness of stimuli in the equivalence classes were not influenced by the functions served by the C stimuli in the equivalence classes.

Two three-member equivalence classes were formed with abstract three-dimensional objects that were observed by touch only by use of matching-to-sample (MTS) trials and the simple-to-complex protocol. Classes emerged immediately for one participant and with a delay in transitivity for the other. For all relational types, reaction time (RT) duration and observing-response (OR) frequency decreased with trial repetition to the very low asymptotes. For the derived relations, error-free decrements implied “learning” without feedback. The highly correlated ORs and RTs suggested that OR frequency accounted for RT duration. Also, information-gathering efficiency per OR increased with trial repetition. Thus, RT duration reflected OR frequency and cognitive processing. Initial RTs were long for baseline trials, much shorter for symmetry probes, intermediate for transitivity probes, and shortest for equivalence probes. More ORs occurred to samples than comparisons and the positive and negative comparisons produced similar OR frequencies. In addition, transitivity performances were predicted by frequency of sample-observing in previously administered baseline and symmetry probes. All findings were based on single-trial data. Finally, observing behavior may also influence the emergence of fluency.

Two experiments used post‐class formation within‐class relational assessment test performances to evaluate whether participants demonstrated preference for certain members of an equivalence class based on the type of relation that existed between class members. In Experiment 1, two 5‐node 7‐member equivalence classes, consisting entirely of nonsense syllables, were established using the simultaneous protocol. Only 1 of the 6 participants in Experiment 1 formed classes. After class formation, the effects of the different relations between stimuli were evaluated using within‐class relational assessment tests, and the 1 participant showed an absolute preference for transitive over equivalence relations, and for baseline over symmetrical relations. Experiment 2 was identical to Experiment 1, except that one of the nonsense syllable stimuli in each class was replaced by a pictorial stimulus. Under these conditions, classes were formed by 5 of 13 participants. During the relational assessment tests, the 5 participants who formed classes demonstrated almost exclusive preferences for transitive relations over equivalence relations and for trained baseline relations over symmetrical relations. Thus, this research demonstrates that the members of equivalence classes are differentially related to each other based on relational type.

This experiment presents a manual-observing procedure as an inexpensive alternative for investigating stimulus control and establishing equivalence classes in a matching-to-sample task (MTS). To illustrate the procedure, we evaluated the effects of different MTS training structures on observing responses and equivalence class formation. Participants had to press a button below each covered sample and comparison stimuli to reveal the stimulus. Four participants were exposed to two different sequences of the many-to-one (MTO) and one-to-many (OTM) procedures, using the manual-observing procedure during training and testing. The results showed that the manual-observing procedure allowed participants to acquire conditional discriminations and form equivalence classes, suggesting that the use of manual-observing responses in an MTS procedure is a useful procedure to evaluate stimulus control in an MTS task.

Three experiments used postclass formation within‐class preference test performances to evaluate the effects of nodal distance on the relatedness of stimuli in equivalence classes. In Experiment 1, two 2‐node four‐member equivalence classes were established using the simultaneous protocol in which all of the baseline relations were trained together, after which all emergent relations probes were presented together. All training and testing was done using match‐to‐sample trials that contained two comparisons. After class formation, the effects of nodal distance were evaluated using within‐class preference tests that contained samples and both comparisons from the same class. These tests yielded inconsistent performances for most participants. Experiment 2 replicated Experiment 1, but a third null comparison was used on all trials during class formation. Thereafter, virtually all of the within‐class probes, for all participants, evoked performances that were consistent with the predicted effects of nodal distance, that is, the selection of comparisons that were nodally closer to the samples. It appears, then, that the establishment of the equivalence classes with a third null comparison induced control by nodal structure of the classes. Experiment 3 demonstrated the generality of these findings with larger classes that contained more nodal separations, that is, three‐node five‐member classes. Emergent‐relations tests conducted immediately after the within‐class tests showed the classes to be intact. Thus, the differential relatedness of stimuli in a class or their interchangeability depended on the content of a test trial: within‐class probes occasioned responding indicative of differential strength among the stimuli in the class, while cross‐class tests occasioned responding indicative of interchangeability of stimuli in the same class.

The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories\"--the \"well generated triangulated categories\"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

Most complex categories observed in real‐world settings consist of perceptually disparate stimuli, such as a picture of a person's face, the person's name as written, and the same name as heard, as well as dimensional variants of some or all of these stimuli. The stimuli function as members of a single partially or fully elaborated generalized equivalence class when they occasion the mutual selection of each other after the establishment of some subset of relations among the stimuli. Indeed, it is these generalized relations among stimuli that enable an individual to respond appropriately to the inevitable flux of natural environments. The present experiments involved procedures for producing both types of generalized equivalence class and for evaluating their retention. Granting the formal and functional similarities that exist between generalized equivalence classes and natural categories, natural kinds, and fuzzy superordinate classes, the variables responsible for the emergence of the former might also account for the emergence of the latter three phenomena. In Experiment 1, After forming an A'—B' class, a B'—C relation was trained and generalization tests were conducted with B'—C, C—B',A'—C, and C'—A. Two of 5 participants passed the tests documenting the formation of A'—B'—C classes. Failures occurred in the A'—C and C—A' tests but not the B'—C and C—B' tests. Failures were also correlated with time between A'—B' class formation and C‐based testing and with the absence of baseline confirmation when training and testing were separated by about one week. Experiment 2 replicated Experiment 1 but presented baseline confirmation probes immediatley prior to testing when training and testing were separated by one week; all participants then formed partially elaborated generalized equivalence classes. In Experiment 3, 5 of 6 participants formed fully elaborated generalized equivalence classes, represented as A'=B'=C'.