We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semi-degenerate congruence-modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive w.r.t. arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence-modular varieties.
We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $\mathcal {V}$ is congruence distributive if and only if the congruence identity $α\cap (β\circ γ\circ β) \subseteq αβ\circ γ\circ αβ\circ γ\dots $ ($k$ factors) holds in $\mathcal {V}$, for some natural number $k$.
Congruence modular and congruence distributive varieties can be characterized by the existence of sequences of Gumm and Jónsson terms, respectively. Such sequences have variable lengths, in general. It is immediate from the above paragraph that there is a variety with Gumm terms but without Jónsson terms. We prove the quite unexpected result that, on the other hand, if some variety has both kinds of terms, then the minimal lengths of the sequences differ at most by $1$. It follows that every $r$-modular congruence distributive variety is $r^2-r+2$-distributive.
Transferring Davey`s Theorem on Annihilators in Bounded Distributive Lattices to Modular Congruence Lattices and Rings
半退化同余模簇的半素代数的同余格满足B. A. Davey的著名的$m$ -Stone有界分配格的特征定理，而且，在这些等价条件下改变基数并不改变它们的有效性。我证明了这一点转移戴维定理从有界分配格，这样的同余格通过一定的格态射和使用的事实，该态射的余域是一个框架。此外，这些等价条件是保持有限的直积这样的代数，和类似的等价满足的元素的半素交换幺环，对偶，由元素的完全剩余格。
We study algebras and varieties where every non-trivial congruence has some class being a non-trivial subuniverse of the algebra in question. Then we focus on algebras where this non-trivial class is a unique non-singleton class of the congruence. In particular, we investigate Rees algebras, quasi-Rees algebras and algebras having the one-block-property. We also present results concerning these properties on quotient algebras. Several examples of such algebras and varieties are included.
We define a relation that describes the ternary commutator for congruence modular varieties. Properties of this relation are used to investigate the theory of the higher commutator for congruence modular varieties.
The {\em reticulation} of an algebra $A$ is a bounded distributive lattice whose prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic to the prime spectrum of congruences of $A$, with its own Stone topology. The reticulation allows algebraic and topological properties to be transferred between the algebra $A$ and bounded distributive lattices, a transfer which is facilitated if we can define a {\em reticulation functor} from a variety containing $A$ to the variety of (bounded) distributive lattices. In this paper, we continue the study of the reticulation of …
We characterize those nilpotent algebras of prime power order and finite type in congruence modular varieties that have infinitely many polynomially inequivalent congruence preserving expansions.
The Lifting Idempotent Property ($LIP$) of ideals in commutative rings inspired the study of Boolean lifting properties in the context of other concrete algebraic structures ($MV$-algebras, commutative l-groups, $BL$-algebras, bounded distributive lattices, residuated lattices,etc.), as well as for some types of universal algebras (C. Muresan and the author defined and studied the Congruence Boolean Lifting Property ($CBLP$) for congruence modular algebras). A lifting ideal of a ring $R$ is an ideal of $R$ fulfilling $LIP$. In a recent paper, Tarizadeh and Sharma obtained new results on liftin…
