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A subset S of an integral domain is called a semidomain if the pairs (S, +) and (S∖{0},⋅) are commutative and cancellative semigroups with identities. The multiplication of S extends to the group of differences 풢(S), turning 풢(S) into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains S for which either s ∈ S or − s ∈ S for every s ∈풢(S)). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains. [ABSTRACT FROM AUTHOR]

Intrathecal synthesis of central nervous system (CNS)-reactive autoantibodies is observed across patients with autoimmune encephalitis (AE), who show multiple residual neurobehavioral deficits and relapses despite immunotherapies. We leveraged two common forms of AE, mediated by leucine-rich glioma inactivated-1 (LGI1) and contactin-associated protein-like 2 (CASPR2) antibodies, as human models to comprehensively reconstruct and profile cerebrospinal fluid (CSF) B cell receptor (BCR) characteristics. We hypothesized that the resultant observations would both inform the observed therapeutic gap and determine the contribution of intrathecal maturation to pathogenic B cell lineages. From the CSF of three patients, 381 cognate-paired IgG BCRs were isolated by cell sorting and scRNA-seq, and 166 expressed as monoclonal antibodies (mAbs). Sixty-two percent of mAbs from singleton BCRs reacted with either LGI1 or CASPR2 and, strikingly, this rose to 100% of cells in clonal groups with ≥4 members. These autoantigen-reactivities were more concentrated within antibody-secreting cells (ASCs) versus B cells (P < 0.0001), and both these cell types were more differentiated than LGI1- and CASPR2-unreactive counterparts. Despite greater differentiation, autoantigen-reactive cells had acquired few mutations intrathecally and showed minimal variation in autoantigen affinities within clonal expansions. Also, limited CSF T cell receptor clonality was observed. In contrast, a comparison of germline-encoded BCRs versus the founder intrathecal clone revealed marked gains in both affinity and mutational distances (P = 0.004 and P < 0.0001, respectively). Taken together, in patients with LGI1 and CASPR2 antibody encephalitis, our results identify CSF as a compartment with a remarkably high frequency of clonally expanded autoantigen-reactive ASCs whose BCR maturity appears dominantly acquired outside the CNS. [ABSTRACT FROM AUTHOR]