The aim of this paper is to study the (a; g) -prolongation of central extensions. We obtain the obstruction theory for (a; g) -prolongations and classify (a; g) -prolongations thanks to low-dimensional cohomology groups of groups.

The main purpose of this paper is to use a new way to extend fuzzy implications I from a generalized sublattice M to a bounded lattice L, such that the extended implications preserve many of the considered proper-ties of fuzzy implications on M. Furthermore, as a special case, we investigate the extension of (S; N) -implications. Results indicate that the extended implications preserve many of the considered properties of (S; N) -implications.

In this paper, we consider a form of the generalized logistic distribution named symmetric extended generalized logistic distribution or extended type III generalized logistic distribution. The distribution is derived by compounding a two-parameter generalized Gumbel distribution with a two-parameter generalized gamma distribution. The cumulative distribution and some properties of this distribution like moments and related statistics are established. Some theorems that characterize the distribution are stated and proved. Estimation of the parameters and an application of the distribution are also presented.

Semantic broadening is one of the important processes of semantic change that has long been studied by experts and especially by historical-philological semanticists. In this process, which occurs over time, the meaning of a word broadens from time period A to time period B and thus, in addition to its previous meaning, the word includes a new concept. Semantic broadening over time leads to a multitude of meanings. A word is used to refer to a new unit of language in terms of its similarity to a new concept, and based on the Prototype theory. As such, in addition to its original meaning, it is also applied in a new sense. This "similarity" has, in many cases, led to confusion between semantic broadening and metaphor, and a group of words that has undergone semantic broadening is considered as metaphors. The present research, in addition to explaining semantic broadening, examines the difference of this process with metaphor, and, in particular, by emphasizing Persian verbs, tries to show that many of the constructs considered as metaphor, are, in fact, instances of semantic broadening that, as a result of this process, have gained multiple meanings. Relying on examples, in particular, examples of Persian verbs, this paper attempted to emphasize the distinction between semantic broadening and metaphor, and to show that, by tradition, in our metaphor, one tends to separate the tenor from the simile and apply it on other contexts. However, in the case of semantic broadening, the tenor is by no means used in different contexts. Words are semantically broadened over time and their semantic scope expands, which is a process quite different from metaphor. As such, the results of this study indicate the subtle distinction that exists between semantic broadening and the process of metaphor, which is often neglected. In metaphor, the "tenor" is derived from the simile, and is used in other sentences instead of the "vehicle"; in a way that the "tenor" adopts the semantic behavior of the “ vehicle” . However, in semantic broadening, an independent process occurs, and what emerges as the process of semantic broadening is the semantic expansion of other units of the simile.

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, ^ (conjunction) and ≡ (equivalence). Moreover, the ⇒ (implication) connective can be derived as A ⇒ B: = (A ^ B) ≡ A. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective Δ (delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.