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We prove our title, and thereby establish the base for a positive solution of Albert and Burris’ problem on the finite axiomatizability of the model companion of the class of all pseudocomplemented semilattices.

The Cantor–Bernstein–Schröder theorem (CBS-theorem for short) of set theory was generalized by Sikorski and Tarski to $$\sigma $$ -complete Boolean algebras. After this, several generalizations of the CBS-theorem, extending the Sikorski–Tarski version to different classes of algebras, have been established. Among these classes there are lattice ordered groups, orthomodular lattices, MV-algebras, residuated lattices, etc. This suggests to consider a common algebraic framework in which the algebraic versions of the CBS-theorem can be formulated. In this work we provide this framework establishing necessary and sufficient conditions for the validity of the theorem. We also show how this abstract framework includes the versions of the CBS-theorem already present in the literature as well as new versions of the theorem extended to other classes such as groups, modules, semigroups, rings, $$*$$ -rings etc.

Every group is the automorphism group of a lattice of order dimension at most 4. We conjecture that the automorphism groups of finite modular lattices of bounded dimension do not represent every finite group. It is shown that ifp is a large prime dividing the order of the automorphism group of a finite modular latticeL then eitherL has high order dimension orM p, the lattice of height 2 and orderp+2, has a cover-preserving embedding inL. We mention a number of open problems.

An extension G ≤ H of lattice-ordered groups is said to be a rigid extension if for each $${h \in H}$$ there exists a $${g \in G}$$ such that h ⊥⊥ = g ⊥⊥. In this paper, we will define rigid extensions and some other generalizations in the context of algebraic frames satisfying the FIP. One of the main results is a characterization of rigid extensions using d-elements of the frame. We also show that a rigid extension between two algebraic frames satisfying the FIP will induce a homeomorphism between their corresponding minimal prime spaces with respect to both the hull-kernel topology and the inverse topology. Moreover, basic open sets map to basic open sets.

In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.

The agreement quasi-order on pairs of (partial) transformations on a set X is defined as follows: $${(f, g) \preceq (h, k)}$$ if whenever f, g are defined and agree, so do h, k. We axiomatize function semigroups and monoids equipped with this quasi-order, thereby providing a generalisation of first projection quasi-ordered $${\cap}$$ -semigroups of functions. As an application, axiomatizations are obtained for groups and inverse semigroups of injective functions equipped with the quasi-order of fix-set inclusion. All axiomatizations are finite.

Các loại được xem xét với p(x, y, z), một phép toán tam phân đơn, hoạt động như một bộ phân biệt cục bộ hoặc bộ phân biệt đối với các phần tử không thể phân rã trực tiếp. Nếu p(x, y, z) là "toàn cục", thì tất cả các tiểu loại đều có cơ sở hữu hạn. Trong trường hợp tổng quát, một liên tục các tiểu loại không có cơ sở hữu hạn được trình bày. Một hình ảnh lý thuyết đồ thị dẫn đến một loại nhóm kết nối các bán nhóm với không trái và không phải. Đối với loại này, một số vấn đề mở được trình bày.

We investigate essential variables of dual operations (or co-operations) in a rather general category-theoretic setting. During this investigation, we collect some results about the essential arities and arity gaps of dual operations in concrete categories. Afterwards, we use duality theory to show that this approach and the obtained results can also be beneficial for questions regarding essential variables of operations on sets.