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We consider systems of inclusions with unknowns and coefficients in multioperations of finite rank. An algorithm for solving such systems by the method of reduction to Boolean equations using superposition representation of multioperations by Boolean space matrices is given. Two methods for solving Boolean equations with many unknowns are described for completeness. The presentation is demonstrated by examples: the representation of the superposition of multioperations by Boolean space matrices; solving a Boolean equation by analytical and numerical methods; and finding solutions to an inclusion with one unknown. The resulting algorithm can be applied to the development of logical inference systems for multioperator logics.

In this paper multioperations defined on a two-element set and their closure operator based on composition operator and the equality predicate branching operator is considered. The composition operator is based on union of sets. The classification of multioperations based on their membership in precomplete sets has been obtained. It is shown that the number of equivalence classes is 129. All types of bases are described and it is proved that the maximum cardinality of a basis is 4.

In algebras of multioperations, unlike algebras of operations, the superassociativity identity does not hold, but only the semi-superassociavity identity is true. For a more detailed study of the identities satisfiable in fixed dimension algebras of multioperations, this work defines the variety to which these algebras belong. In particular, among these identities defining a variety, an identity is introduced that similar to the Dedekind relation for binary relations. From the introduced identities, some consequences are derived that satisfiable in the fixed dimension algebras of multioperations. Note that the variety is defined in a language whose symbols are interpreted by the superposition metaoperations, the first argument permissibility, and constant projection metaoperations for each argument and the zero multioperation. In this language, the terms are the intersection meta-operations, the permissibility by any argument, the full multioperation, and the inclusion multioperation. Another interesting task is studying quasiidentities satisfiable in the fixed dimension algebras of multioperations.

Multioperations are operations from a finite set A to set of all subsets of A. The usual composition operator leads to a continuum of closed sets. Therefore, the research of closure operators, which contain composition and other operations becomes necessary. In the paper, the closure of multioperations that can be obtained using the operations of adding dummy variables, identifying variables, composition operator, and operator with the equality predicate branching is studied. We obtain eleven precomplete closed classes of multioperations of rank 2 and prove the completeness criterion. The diagram of inclusions for one of the precomplete class is presented.

The construction of Galois theory for the algebras of operations and relations is a popular topic for investigation. It finds numerous applications in both algebra and discrete mathematics – especially for the perfect Galois connection, since if such a connection is established for the sets of all subalgebras of some algebra then the algebraic closure in this algebra coincides with the Galois closure, and this is an efficient tool for solving many algebraic problems. The perfect Galois connection is well known for clones and co-clones, as well as for some algebras of operations, for example, for clones and superclones. In all these cases, infinite algebras are considered. In this article we study Galois theory for finite algebras of operations and multioperations with fixed rank and arbitrary dimension. We find the necessary and sufficient conditions on the dimension of algebras so that the Galois connection between the lattices of subalgebras of algebras of operations and algebras of multioperations of rank 2 is perfect. The problem of finding the necessary and sufficient conditions for the existence of a perfect Galois connection between lattices of algebras of operations and algebras of multioperations of arbitrary fixed rank is posed.

The idea of designing multioperation mechanisms capable of performing different tasks has gained prominence in the last years. These mechanisms, commonly called reconfigurable mechanisms, have the ability to change their configuration. At present, this type of mechanisms is capturing the attention of design engineers because of their great potential in many industrial applications. In this paper, the basis for the development of a methodology intended for the analysis and design of multioperational parallel manipulators is presented. First, the structural synthesis of 6 degree-of-freedom (dof) kinematic chains that can form a 6 dof manipulator is established. Next, a general purpose approach for non-redundant parallel manipulators (PM) will be presented. This procedure enables obtaining the Jacobian matrices of any 6 dof or low-mobility PM whose kinematic chains belong to the library of chains derived from the structural synthesis. To demonstrate the versatility of the procedure, it will be applied to three PM: the first one, a 6 dof PM, the second one, a reconfigurable 6 dof PM, and finally, a low-mobility PM. [ABSTRACT FROM AUTHOR]

Multioperations are primitives of parallel computation by which threads perform reductions, e.g., additions, on values provided by multiple threads into a single value in a constant number of steps. Multiprefixes resemble multioperations, but return to each participating thread a cumulative ordered reduction of all preceding values. Algorithmically, multioperations and multiprefixes can speed up parallel programs by a logarithmic factor over their single operation counterparts. In this paper, we introduce architectural techniques for realizing multioperations and multiprefixes in so-called Thick Control Flow (TCF) processors. A thick control flow is a computational construct that bundles homogeneous threads following the same control path into a data parallel entity. Our proposed processors optimized for executing TCFs feature a frontend-backend structure with low-latency processing of TCF-common computations and high-throughput execution of data parallel parts. Our solution relies on step caches and equally sized multioperation scratchpads, while on the memory side, we make use of active memory modules. The idea is to compute partial results in backend units to reduce the traffic to the referred shared memory location. The final multioperation result is then computed in the active memory unit of the target memory module. Multiprefixes use an additional phase where the final results are computed with a help of backend-wise prefixes. According to our evaluation, the proposed techniques indeed speed up certain N data element algorithms by a log N factor with reasonable hardware costs. [ABSTRACT FROM AUTHOR]

We study clones (closed sets of operations that contain projections) and superclones on finite sets. According to A. I. Mal'tsev a clone may be considered as an algebra. If we replace algebra universe with a set of multioperations and add the operation of simplest equation solvability then we will obtain an algebra called a superclone. The paper establishes Galois connection between clones and superclones. [ABSTRACT FROM AUTHOR]