Algebra Universalis最新文献

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term (theta )). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the (theta ) appearing in their syntactic characterisation). Restricting again to the case of monoids, a different choice of (theta ) leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.

Each strongly minimal Steiner k-system (M, R) (where is R is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if k is a prime-power. We show this coordinatization is never definable in (M, R) and the strongly minimal Steiner k-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction, if k is a prime power, in each (2, k)-variety of quasigroups (Definition 3.10) there is a strongly minimal quasigroup that interprets a Steiner k-system.

We consider ordered universal algebras and give a construction of a join-completion for them using so-called (mathscr {D})-ideals. We show that this construction has a universal property that induces a reflector from a certain category of ordered algebras to the category of sup-algebras. Our results generalize several earlier known results about different ordered structures.

Stone space partitions ({X_{p}mid pin P}) satisfying conditions like (overline{X_{p}}=bigcup _{qleqslant p}X_{q}) for all (pin P), where P is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of (omega )-categorical structures. A key concept for studying such partitions is that of a p-trim open set which meets precisely those (X_{q}) for which (qgeqslant p); for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition ({X_{r}mid rin [0,1]}) of the Cantor set such that (overline{X_{r}}=bigcup _{sleqslant r}X_{s}text { for all }rin [0,1]).

Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset X in an algebraic lattice P generates a closure system (langle X rangle _P). The closure system (langle Y rangle _P) generated by the patch closure Y of X is the patch closure of (langle X rangle _P). If X is contained in the set of nontrivial prime elements of P then (langle X rangle _P) is a frame and is a coherent algebraic frame if X is patch closed in P. Conversely, if the algebraic lattice P is coherent then its set of nontrivial prime elements is patch closed.

In this article we study the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.

The article proves topological representations for some classes of double Boolean algebras (dBas). In particular, representation theorems characterising fully contextual and pure dBas are obtained. Duality results for fully contextual and pure dBas are also established.

We show that any ordered group satisfying the identity ([x_1^{k_1}, ldots , x_n^{k_n}] = e) must be weakly abelian and that when (x_i not = x_1) for (2 le i le n), (ell )-groups satisfying the identity ([x_1^n, ldots , x_k^n] = e) also satisfy the identity ((x vee e)^{y^n} le (x vee e)^2). These results are used to study the structure of (ell )-groups satisfying identities of the form ([x_1^{k_1}, x_2^{k_2}, x_3^{k_3}] = e).

Let S be a multiplicatively idempotent congruence-simple semiring. We show that (|S|=2) if S has a multiplicatively absorbing element. We also prove that if S is finite then either (|S|=2) or (Scong {{,textrm{End},}}(L)) or (S^{op}cong {{,textrm{End},}}(L)) where L is the 2-element semilattice. It seems to be an open question, whether S can be infinite at all.